Here is a tweaked version of the Constitution class where it has multiple warp cores and Dual impulse engines. I also gave it some other modifications to the Impulse engines such as giving it a boost of power rather than increasing the speed. I was looking at how the ship translated from screen to Spacedock and soon figured out that it was under powered for a fight such as those I will note in the text below. With a little tweaking this was what I came up with for the ship and how it would function in combat in the episode. Most of the ship remained the same as it was from spacedock but the changes I made are just thinking.
The single impulse engine is not powerful enough even supplemented with auxiliary and emergency power to operate the ship during a battle scenario (such as the episodes of Journey to Babel and Elaan of Troyius (I think that is how the episode was spelled)) with the warp drive power knocked out. With the single impulse engine here is not enough power even supplemented by the auxiliary and emergency power is lacking giving not enough power to shield the ship.
Original version of power - Available Power (Impulse power): 23
Original version of power - Available Power (Impulse and Auxiliary power): 43
Original version of power - Available Power (Warp power): 280
Original version of power - Available Power (Impulse and Warp power): 303
Original version of power - Available Power (Impulse, Warp and Auxiliary power): 323
Original version of power - Available Power (Impulse, Auxiliary and Emergency power): 83
Original version of power - Available Power (Impulse, Warp, Auxiliary and Emergency power): 384
This leaves not power for shields. Even if I employ a Janeway tactic of over running the engines by 10% would give only a small boost of two power for the ship helpful but not enough. Our house rules no more than 15% and at 20% engines are taking too much damage and damaged. Each round engines take one damage for ever round and double damage one for every 5% above 5%. So at 20 percent the ship is taking four damage per round and after four rounds of that the engines are done for irreparable outside of a dry dock as they will need to be replaced for the Constitution class.
However there are two or four vent openings on the stern of the saucers impulse manifold suggesting four engines and then the ship is far too powerful with four engines. (This depend on the images and the drawing you look at. I stay with the two myself.) So I gave it a power boost with 8 more points of power at a cost of 5 SUs for the gain. However, with twin Impulse Engines like the refit this gives the ship minimal shields 25% (protection 100), moving .1c or using thrusters and one to two phaser shot or pivoting firing two or three shots from the forward Photon Torpedo launcher. I lowered the warp power to compensate for the increased power from the impulse engines a little.
The optional multiple warp core style is unrealistic except still allows a ship to be battlefield functional even if one or two cores are knocked out it is not like the Klingon ship with twin cores nevertheless with similarities.
A tweaked out version of the Constitution Class
Class and Type: Constitution Class Explorer
Commissioning Date: 2245, underwent substantial upgrades in 2258, and 2263
Environmental Systems
Basic Life Support [11 Power/round] <24> …….. (off line) [0 power]
Reserve Life Support [6 Power/round] <12> …… (operating) [6 power]
Emergency Life Support (24 emergency-shelters) <12>
Gravity [3 Power/round] <6> ……………………. (operating) [3 power]
Consumable: two years worth <24>
Food Processors Mark IV [4 Power/round] <18> .. (off line) [0 power]
Industrial Fabrication Units Mark VII [5 power/round] <18> … (off line) [0 power]
Medical Facilities: 6 (+1) [6 Power/round] <30> .. (operating) [6 power]
Recreation Facilities: 8 [8 Power/round] <48> …. (off line) [0 power]
Personal Transport: Turbolifts, Jefferies tubes [2 Power/round] <18> … (operating) [2 power]
Fire Suppression System [1 Power/round when active] <6> … (standby) [0 power]
Cargo hold: 15,000 cubic meters <1>
Locations: six locations throughout ship
Escape Pods <7>
Number: 140
Capacity: 4 persons per pod
Propulsion Systems
Warp Drive Nacelles: Mark 4.68 <53>……......... (off line) [0 power]
Speed: 4.0/6.0/8.0 (TOS) [1 Power/.2 warp speed]
PIS: Class H (12 hours of Maximum warp) <16>
Impulse Engine: double tweaked Type 5A (.5c/.75c) [5/7 Power/round] <28> … (speed .1 of light speed) [1 power]
Location: Port aft of saucer section
Impulse Engine: double tweaked Type 5A (.5c/.75c) [5/7 Power/round] <28> … (speed .1 of light speed) [1 power]
Location: Starboard aft of saucer section
Reaction Control System (.025c) [2 Power/ round when in use] <6> … (operating) [2 power]
Power Systems
Warp Engine Type: Mark VI (generates 258 Power/round) <71>
(Optional) Warp Engine Type: six Mark I (generates 43 Power/round) <25 x 6 = 150>
Locations: Engineering hull
Impulse Engine[s]: Port Type 5A (generate 33 power/engine/round)
Impulse Engine[s]: Starboard Type 5A (generate 33 power/engine/round)
Auxiliary Power: 4 reactors (generate 5 Power/reactor/round) <12>
Emergency Power: Type D (generates 40 Power/round) <40>
EPS: Standard Power flow, +120 Power transfer/round <42>
Available Power (Impulse power): 66
Available Power (Impulse and Auxiliary power): 86
Available Power (Warp power): 258
Available Power (Impulse and Warp power): 324
Available Power (Impulse, Warp and Auxiliary power): 344
Available Power (Impulse, Auxiliary and Emergency power): 126
Available Power (Impulse, Warp, Auxiliary and Emergency power): 384
Operations systems
Bridge: Saucer dorsal <24>
Auxiliary Control Room: Engineering hull <12>
Saucer Separation: Saucer separation [10 Power/round] <3>
Computers
Cores: [5 Power/round] <12> …………………. (operating) [5 power]
Cores: [5 Power/round] <12> ………………….. (off line) [0 power
Uprating: Class Alpha (+1) [1 Power/computer round] <4>.. (operating for one core) [1 power]
ODN <18>
Navigational Deflector [6 Power/round] <18> … (off line but on standby) [0 power]
Range: 8/15,000/40,000/125,000
Accuracy: 6/7/9/12
Location: Forward engineering hull
Sensor Systems
Long-range Sensors [5 Power/round] <37>……. (off line) [0 power]
Range package: Mark VIII (Accuracy 4/5/8/11)
High Resolution: 5 Light-years (0.5/0.6 - 1.0/1.1 - 3.7/3.8 - 5.0)
Low Resolution: 15 Light-years (1.0/1.1 - 4.0/4.1 - 12.0/12.1 - 15.0)
Strength Package: Class 6 (strength 6)
Gain Package: Class alpha (+1)
Coverage: standard
Lateral Sensors [5 Power/round] <15> …………(Operating) [5 power]
Strength Package: Class 6 (strength 6)
Gain Package: Class alpha (+1)
Coverage: standard
Navigational Sensors: [5 Power/round] <14> …. (off line) [0 power]
Strength Package: Class 6 (strength 6)
Gain Package: Class alpha (+1)
Probes: 30 <3>
Sensor Skill: 4
Flight Control Systems
Autopilot: Shipboard systems (flight control) 2, Coordination 1 [1 Power/round in use] <1> … (on standby) [0 power]
Navigational Computer
Main: Class 2 (+1) [1 Power/round] <2> ………. (on standby) [0 power]
Backups: two <2>
Inertial Damping Field
Main <24>……………………………………… (operating) [6 power]
Strength: 8 [3 Power/round]
Number: 2
Backup <6> ……………………………………. (on standby) [0 power]
Strength: 5 [2 Power/round]
Number: 2
Attitude Control [2 Power/round] <2> ………… (operating) [2 power]
Communications Systems ……………………… (operating) [3 power]
Type: Mark V [3 Power/round] <22>
Strength: 5
Security: -3 (Type A uprating)
Basic Uprating: Type 1 (+1)
Emergency Communications: [2 Power/round] <1>
Emitter: Class Alpha [3 Power/Strength used/round] <3> … (off line) [0 power]
Accuracy: 5/6/8/11
Location: Hanger bay
Transporters
Type: Personnel [6 Power/use] <14>………………………. (on standby) [0 power]
Pads: 6
Emitter/Receiver Array: Personnel Mark 6 (26,000 km range)
Energizing/Transition Coils: Class E (strength 5)
Number and location: one in saucer
Type: Emergency [7 Power/round] <15 x 2 = 30> ………… (off line) [0 power]
Pads: 22
Emitter/Receiver Array: Emergency Mark 4 (13,000 km range)
Energizing/Transition Coils: Class E (strength 5)
Number and location: one in saucer, one in engineering
Type: Cargo [2 Power/round] <9 x 2 = 18>………………… (off line) [0 power]
Pads: 200 kg
Emitter/Receiver Array: Cargo Mark 4 (26,000 km range)
Energizing/Transition Coils: Class E (strength 5)
Number and location: Two in engineering section
Security Systems Rating: 3 <12>
Anti-Intruder System: [1 Power/round] <6> ………………. (on standby) [0 power]
Internal Force Field [1 Power/3 Strength] <6> ……………. (on standby) [0 power]
Science Systems Rating: 3 (+2) [3 Power/round] <21> …… (on standby) [0 power]
Specialized Systems: 3 Planetary Analysis, Botanical Analysis, Stellar Mapping <15>
Laboratories: 14 <2>
Tactical Systems ……………………………………………(34 power available for fighting (two good phaser shots or two or four photon torpedoes (if you play with the power here and there dropping something here and there. I like torpedoes but phasers are easier.)))
Forward Phaser Bank <24> ………………………………. (operating and charged) [28 power]
Type VII
Damage: 140 [14 Power]
Number of Emitters: 120 (up to 3 shots per round)
Targeting Systems: Class Beta (Accuracy 4/5/7/10)
Range: 10/30,000/100,000/300,000
Location: Forward Ventral
Firing arc: 240 degrees forward ventral
Firing Modes: Standard, Continuous, Pulse, Wide-beam
Port and Starboard Phaser Banks <22 x 2 = 44>…………. (off line) [0 power]
Type VII
Damage: 140 [14 Power]
Number of Emitters: 120 (up to 3 shots per round)
Targeting Systems: Class Beta (Accuracy 4/5/7/10)
Range: 10/30,000/100,000/300,000
Location: one port Dorsal on the saucer, one starboard on the saucer and aft dorsal
Firing arc: 120 degrees aft dorsal
Firing Modes: Standard, Continuous, Pulse, Wide-beam
Phaser Control rooms <6>
Photon Torpedo Launchers <15> ……………………… (off line) [0 power]
Standard Load: Type I Photon Torpedo (160 damage)
Spread: 6
Range: 15/100,000/400,000/750,000
Targeting Systems: Class Beta (Accuracy 4/5/7/10)
Power: [20 + 5 per torpedo fired]
Forward Ventral Saucer
Firing Arc: Forward, but are self-guided
Torpedoes carried: 40 <4>
Torpedo control room <6>
Shields (Forward, Aft, Port, Starboard) <34 x 4 = 136> … (operating 25%) [40 power]
Shield Generator: Class 2 (protection 400) [40 Power/shield/round] … (protection 100)
Shield grid: Type B (33% increase to 533 Protection)
Subspace field Distortion Amplifiers: Class Gamma (Threshold 133)
Recharging System: Class 0 (60 seconds)
Backup Shield Generators: none
Auto-Destruct System <6>
Auxiliary Spacecraft systems
Shuttlebay(s): Capacity for 12 Size worth of ships <24>
Standard Complement: six shuttlecraft
Location(s): aft secondary hull