I won't compete with selek for the length of the post nor the number of suggestions
, but here are some quick ideas :
First, in what era will this happen ? If this is post DW, then they might manage to ask free passage through the Romulan Empire indeed (or they could try to sneak through, since the Romulans have lost a lot of ships during the war and some areas may not be as heavily guarded). OTOH, if this happens before the DW, things will be a lot trickier for the Crew.
One solution would be for them to try to go along the border of the Empire. Thus, the journey will be much longer, and still dangerous since the ship would find itself between the Romulans and various powers - maybe at war with the Romulans.
Also, it might be interesting to cut their communications with the Federation, or at least put them at risk of being detected.
Plus, they may have to spend some time to repair their ship (and stop on a lot of unknown planets).
Apart from these points, good idea, Cmdr. Matt 
I may borrow it for my games.
"The main difference between Trekkies and Manchester United fans is that Trekkies never trashed a train carriage. So why are the Trekkies the social outcasts?"
Terry Pratchett