Mayo is a suspended emulsion of oil and vinegar using egg as a binder. The trick is to mix the oil into the egg in as small of droplets as you can so the egg can suspend them and not break down and split. A whisk would help with this, but I don't think it is necessary. It might be labor some and require more skill, but you could probably do it with other pronged tools.
The vinegar in the mayo would help break down the proteins in the rabbit and make them softer, but heat is a great way to cause a mayo to split. The sauce would be ruined by cooking it this way.
Ideally you would marinate the meat first, cook the ingredients, and then serve the sauce on top or to the side.