After seeing you questions, it really would be more viable for you to take a tin of spam with you! Getting written permission to hunt on private land can be a bit of a nightmare, getting permission to wild camp on someone's land and hunt very very rare. The idea of wild camping is to just camp without permission ie arrive late leave early, so if you add armed trespass, ie hunting with out permission to this as well then, you maybe looking at jail time because it is illegal. Also you'll need a knife for the gutting and skinning so they'll have you for that as well. My advice to you is first, find somewhere to shoot your catapult, then practice every day for however long it takes to get really proficient. Then join a good gun club, air rifle club, buy some quality quality gear and get some qualified instruction and when you can hit a 2p at 35 yards every time, then start to look for hunting permission. Once you have hunting permission and you have built up a good working relationship with the land owner then ask about camping on remote spots on his land. After the landowner has got used to this and trusts you not to burn everything down, you could mention that you are thinking of taking your gun along as you've seen rabbits and pigeons from you camp and could you shoot them. Bingo you have made the dream you want to come true.
Trespass and shooting do not go together, hearing the cocking of several automatic rifles and the words " Step away from the vehicle, put your weapon down, get down on your knees and put your hands on the back of your head". Is not the thing you want to hear too often in your life. I know from bitter experience.
I'm really trying not to be too negative about this for you, it can be done, it takes lots of time that is all. I have three places that afford me this opportunity, it's great to go, hunt, camp and cook something you have dispatched your self, but after spending a lifetime hunting and doing pest control and 10 years as an Explorer scout leader, sometimes a tin of bacon grill and a warm bed are what's really needed.
Si