As I understand you have in mind that the restaurant would "serve" live animals alongside many vegan dishes? It wouldn't be just a place where you can get a live animal and nothing else right?
The idea is good, but it's practically a bit complicated in its original form. Such a restaurant would have to be an animal shelter at the same time, so you'd need a big space for this, somewhere on the outskirts of the city, etc. This is because vegans care about animals a lot, so the animals kept on the premises of the restaurant, waiting to be bought should preferably be kept in the best conditions possible. They should be well taken care of because if no one buys an animal for long, it can't be kept in a cage or in any bad conditions, this would upset the customers a lot.
So an option to place the animal in an animal sanctuary by paying for it is a tricky one, cause preferably the animals should directly go where they'd be best taken care of, and halting them in a restaurant only seems like an unnecessary in-between stop that should be avoided. I think some vegans would see it that way - that the restaurant is making business at the expense of the animals.
Arguably the only better scenario for rescued farm animals rather than being in a sanctuary is if someone "adopts" them into a pet. So I propose that's what the restaurant would offer - to adopt animals as pets. The more animals the sanctuary has the more resources it needs to sustain itself, so if good people adopt some of their animals as pets they'd be happy. The adoption might be free or might be for a price.
I also propose not using the approach of putting animals on the menu and buying them in a similar manner you buy food. This is also likely to cause resentment for some. A restaurant could perhaps have the "adoption menu" with pictures, names, and personalized descriptions of the animals, but without prices. You could talk with a staff member if you considered adopting some animal seen on the "menu".