LOL, and I am sure now that I have done it wrong, it will stick and I won't be able to remember the right way to spell her name
The limited resources, to me is the crux of this: While Klingon death Rituals mean they should be able to do an autopsy without much of a problem (since the dead body is just a shell) Without a forensic suite, an autopsy is going to be impossible, let allone admisible in court, if it wasn't done under clinical conditions !
But yes a whodunnit with Klingons could quickly degenerate into a farce, and a slaughter fest! On the flip side Batleth woulds are easier to heal that phaser burns (on setting 8 ! ) So it needs to be handled with as much subtlety as possible!
It could well be that the Starfleet presence is what will help it to be solved: because the Klingons being carefull arround them, and they will investigate the crime in a way that the Klingons wouldn't nor expected: Like Starfleet! Of course the flip side to that is lots of attempted assasinations!
Could you do ... both? Perhaps the Federation character advocates the use of technology to solve the crime, whereas one of the Klingon guests is an ex-law enforcement type who believes that the right approach is to sweat all the suspects and get all the facts. You could build dramatic tension out of the very difference between the two genres; each advocate is going to be certain his approach is correct. For a twist, you could even have two Klingon guests advocating the two different approachs, and then set the tone based on which one the PCs seem to favor.
whereas one of the Klingon guests is an ex-law enforcement type
Hmmmm... This leads to an interesting question on Klingon culture- do the Klingons have a dedicated police force? What form does Klingon law enforcement take? What rights are accorded the accused?
We do know that Klingon society is more or less based on Feudal lines, with the House as the basic unit of societal structure. It seems that rather than a standing military, each House supplies its own ships and personnel. It would make sense that law enforcement would follow this pattern. Each Houre could have an equivalent to the mediæval Engish High Sherrif, who controls a force of armsmen. There would likely be one or more investigative types on staff. We got a taste of Klingon jurisprudence in Star Trek VI, so we know s bit about how their courts work...
My basic assumption is that any race as fractious as the Klingons would have to learn, eventually, the value of interhouse cooperation. Otherwise, any crime (or even slight) would result in an escalating blood feud that would at the least debilitate both houses. Doubtless this is how things worked in the earliest glimmerings of Klingon civilization. But in order to develop a conquering empire, at some point, they would have to devise a mechanism for keeping their aggression focused outwards.
Some possibilities:
Law enforcement is essentially a "house" of its own, and it draws from interested individuals of other houses -- who must renounce their heritage and embrace a new one. The law enforcment "house" earns honor by solving crimes and punishing the perpetrators; they supply investigators, judges, interrogators -- whatever is needed. In order to function (and survive), they would have to cultivate a relentless degree of unimpeachability. They punish -- relentlessly -- any of their own who cannot uphold this code.
Law enforcment is a cooperative effort. Members retain their own house affiliations, which occasionally leads to infighting, political and otherwise, but for the most part they are effective enough to do their job.
This mechanism would chiefly handle those crimes that cross house boundaries. Each house would retain its own experts to handle in-house matters, and there could easily be jurisdictional squabbling between the umbrella unit, whatever it is, and the house investigators. In house officials would tend to regard the umbrella agency as self-important buttinskys (like some local police here regard the FBI), whereas the umbrella agency would regard the house specialists as know-nothing bumpkins (as some FBI agents regard local police departments).
Justice and vengeance are kissin' cousins, but I suspect the Klingon notion of justice would place heavy emphasis on whether the suspect was actually guilty, and less on some of the rights we consider important, such as the right to privacy. I doubt they would have anything like our Fourth or Fifth Amendments, for example. That's not to say they'd pick a victim and railroad him, because such an act would be dishonorable. It does mean they'd be less concerned with individual rights where those might trump the need to determine who was guilty.
I would suspect that your second variant for inter-House law enforcement - that it is a cooperative measure between Houses - is closer to "reality."
As I see it, the House is the basic societal unit, and even the High Council is essentially made up of House representatives. Even the Imperial House is just the first among not-really-equals in this matter. If there is a Klingon version of Interpol, it's almost certainly as you describe it, "a cooperative effort. Members retain their own house affiliations, which occasionally leads to infighting, political and otherwise, but for the most part they are effective enough to do their job."
As for rights, we have to look at Klingon society as a whole to get a handle on things. Klingons under suspicion may have fewer rights than enjoyed by citizens of Western nations which base their legal systems on English Common Law, but they probably also have rights we don't, like the right to challenge an accuser to a duel to the death to decide the matter. Further, the rights you have almost certainly increase with your rank and influence (think Honour points, similar to ICON's Romulan Glory points).