My first FAC rifles were air rifles - mostly Theoben Rapids. If you can find one, the absolute best calibre for rabbits and other vermin is .20 running at around 25ft/lbs, preferably with a BTAS regulator. Incredibly accurate and efficient out to 70+ yards. Pellets can be hard to find but have the advantage of being orderable by post.
Otherwise, BAS Super 10s are worth a look too, particularly if you can find one that has been blueprinted by John Bowkett in Solihull. There are always some FAC bargains to be had on the AirgunBBS.com sales pages.
I wouldn't bother looking at any FAC rifle that wasn't fed via a buddy bottle, as the low shot count from standard tubes PCPs makes them a pain.
Adam.