if you are looking for a dog that is a hunter with a big heart you should look into getting a labrador retriever they have big mouths that can hold a good size animal and they are fast they also will love you to death
It depends on what you want the dog to do...Pointers will show you the game and flush it...retriever will retrieve the game. Spaniels will also retrieve. Setters if I am not mistakes show and flush like pointers. Look at all the sporting breeds and figure out what their job will be..
OK, you need a poodle! I know it sounds silly but poodles are amazing hunting dogs! they're the second most intelligent dog so they're easy to train and also they were bred to me hunters and water retrievers, they are also very quick and agile and should be able to easily catch a rabbit!
You need a pointer or a setter. Retrievers may be tempted to "retrieve" your bird of prey, where as the job of a pointer is to "show" where the bird where the game is. If you need a dog that will point and retrieve, for instance you're planning to work with bird and gun on different occasions then a spaniel is your best bet as they can stop and point to show where game is as well as retrieve and flush. Extra training will be required though. Or there are "sprionters" which are springers crosses with pointers, of course as with all cross breeds there is no guaranty the dogs behaviour will be 50% mum 50% dad.
Hope this helps