Romulan Ale - This can be found by (re)playing Romulan missions. It is sometimes found in the crates scattered around the maps, along with the standard random hypos and such. Perhaps it drops more often as well, but I'm not sure.
Romulan ale is also found through 'Confiscate Contraband from Crew' duty officer assignment, though it is only one of several possible 'confiscated' items.
You can also get your bartender on your ship interior to do the 'Special Reserve' assignment for you, which can yield romulan ale on critical success.
Dosi Rotgut - The primary way of getting this is through an 'Experimentation with tulaberry based Cocktails' duty officer assignment from your 'personal' list. However, you must complete the tulaberry biochemical chain to access it.
You can also get Rotgut from Confiscate Contraband assignments.
There is absolutely no reason to get the Starbase Bartender since it is the same Bartender as what is available on your ship. If you have started it, then it is going to be awhile before you complete it.