Nah. Don't have an FAC. Besides in my county for fox, you have to have CF, usually .223 min, even on 1st grant.
For the numbers of rabbits I shoot, I'd probably use a .223 as an all round gun.
If numbers became an issue, then yep I'd take a HMR as a 2nd gun over a .22 any day, unless I was shooting near housing at night. A few cracked cases and for some it becomes the devils spawn. Personally I don't see it. The superior range make the HMR a very nice little tool for small pest control. Every calibre has an achilles heal and .22 is far from perfect - low power, limited range and a tend