Rabbits are small targets, so you need a really accurate rifle. Many semiauto rifles like the Ruger Mini 14 are not at all very accurate. A good bolt action rifle would be better.
Recommendations for the .243 Winchester are way over kill for rabbits. I have seen even the largest Jackrabbits literally explode when hit by the .243. It would be ideal for the bigger game that you mention, though.
If you have the ability to handload your own ammo, a rifle in either the 17 Remington Fireball or 221 Remington Fireball are about the ideal centerfire round for rabbits. Both use little powder, and small .17 or .22 caliber bullets. The .20 caliber .204 Ruger is another option, although even it is a bit too powerful. A .22-250 would also devastate a rabbit like a .243, unless you handloaded it down in performance.
Are you absolutely sure that rimfire ammo is forbidden for small game too? That sort of regulation makes no sense at all. Centerfire ammo has far greater range, and is thus much more dangerous than rimfire ammo. That hunting is even allowed in your National Parks is exceptional. No hunting of any kind is allowed in any of our National Parks or National Monuments. And PETA is now trying to get fishing banned from them also.
Here is a review of a link to my Rabbit rifle:
http://www.gunblast.com/Savage-93R17.htmRuger and CZ also make good rifles in the .17 HMR caliber.
And here is a review of a CZ rifle in .221 Fireball:
http://www.gunblast.com/CZ-527.htmIf you have to limit yourself to only one gun, then perhaps you could compromise with the .223 Remington. But get a Bolt action, and not an auto rifle.
