I've not been to Bluebird, but have been to the other three.
Just so you know, the Publican is by far the priciest of these options, especially if you plan to eat oysters or some nice ham. At the same time, though, the Publican is the only one of the three restaurants where everything I tasted was good, if not great. And, if you want oysters, this is the spot. Granted, I visited the Publican early in the evening during their first week of service - - and Paul Kahan was there making sure everything was going well. Two reasons we haven't been back - - subsequent negative LTH posts and the feeling that the food is too pricey for what it is.
So, if I had money to burn or someone else was paying, I'd go to the Publican. No question.
I really want to like Paramount Room, though, and have given them a few tries. If you like fried fish, they do an amazing job. And, I really like the tempura green beans. The steak tartare is better than okay (but not as good as the Publican's). But, twice the burger has been unacceptable, and the fries are consistently poor. The beer list is solid and I like the space. In fact, if I were flying solo, I'd definitely pick Paramount Room, as I'd find it most comfortable and approachable dining alone.
I've only been to the Bristol once, and expect to return before too long. I like what they're doing there, with locally-focused changing menu items. And, the chef seems like a really good guy. The servers handled the few problems appropriately and with finesse. If there hadn't been any problems, I may have visited again by now. The Bristol, IIRC, is pricier than Paramount Room.
So, to sum it up, if I were alone, I'd go to Paramount Room. If I had company and didn't mind spending some cash, I'd go to the Publican. If I were a bit cost-conscious and had company, I'd go to the Bristol.
Sounds like a fun day, in any event. You can't go wrong.
Ronna