I agree with the two posters immediately above.
To be clear, I don't want my spaniels (English Cockers) to mess around with bunnies scent or follow them. I want them to be totally focused on grouse, w/c, roosters, prairie grouse, sharpies and other upland birds.
But over across the pond in the British Isles, a "rough shooting" day implies that the spaniels (ESS or ECS) search out and flush pheasant, European w/c, grouse, rabbits, hare, pigeons or sit on the bank and wait for ducks to come in. In other words, to hunt whatever may be around.
In this country we usually are more specialized. If the OP is after rabbits, get a beagle; I love to listen to the little hounds. We had beagles when I was a kid and had a ton of fun behind the hounds. And if the OP is really after "whatever is around", the one of the spaniel breeds will work well in that role.