I'm assuming you don't hold a shot gun license? If I was you I'd stick to an air rifle for rabbits. Going the FAC route just to shoot rabbits seems a bit OTT to me (given the process and financial aspect - application fee, cost of rifle/ammo safe, rifle, scope, ammo...) and I doubt you'll be granted one just for that anyway.
Obviously with the air rifle you can shoot pigeons both on the roost and on the ground. I don't know anyone who goes out specifically after woodies with a rimfire especially given that they can only shoot them on the ground. Shotgun would be ideal for decoying woodies,