When we were young, my sister and I had mentors and studied enough to get our falconry licenses. We didn't take the exam because we didn't have any possiblity of having a mews. However, I can tell you that falconry is an exacting, time-consuming, and specialized sport. You don't just have a bird and then let it patrol the area. You carry the bird hooded and fly it when you're ready. The bird isn't going to kill a dozen rabbits for you in a day - it may not kill any. When it misses its mark, and sometimes when it hits it, the bird can break feathers, some of which will need repair (you'll learn how to do that) before it can hunt again. You will have to monitor it's weight, to the gram, so you know when it is not too full from feeding or too weak from hunger to hunt well. When it does hunt, you will have to be on hand to dispatch the prey animal quickly if the bird has not done so, as concern for prey animals is a part of falconry.
Further, at least in our state, Utah, the only birds you can get as a novice are either Kestrels or Red-Tail Hawks. Kestrels hunt grasshoppers, beetles, mice and occasionally small birds. Rabbits are at the top end of prey size for a Red-Tail Hawk. This bird will much more commonly catch mice, squirrels, snakes, and other smaller creatures. Also, at least in Utah, you have to learn how to catch and train your own bird.
I guess what I'm getting at is this - if you want to hunt, buy a gun or bow or slingshot and practice with it. If you want to control rodents in your nursery get some cats or a good, small terrier and let nature take its course. (My Cairn is absolutely spooky with rodents.) If you want to be involved in learning all about falcons and hawks in minute detail, and you want to spend a great deal of time working extensively with them, with hunting purely as an incidental occurrence, get a falconry license.