16 gauge is rated for 17A; 24 gauge is rated for 3A. Given that, as you say, the rated current for the doorbell is 1A, I don't understand why 16 gauge is necessary; if you're concerned about shorts, consider that 16A/16V would more likely burn out the transformer before doing anything to the wires because the power density is larger there. Just to be safe, I'd put 3A fuse inline and use Cat5 without reservations.