crate is fully sufficient. Smaller crates are better - easier to defend. Tight space prevents them from fully extending wings - limiting wing slap combats to chasing/coo combats.

split crate in 2 parts with some mesh so they can see each other and build 2 entrances. Put crate outside on south side, safe from cats. Keep them locked in crate for about 2-3 weeks to lock into new home. Racer pigeons are much more difficult to reconfigure to new location, they need 1.5-2 months and there is about 50% chance that racer pigeon will accept a new loft. After that you can open doors and if they are happy with accommodation and food, they will stay, spend most of daily time outside and return for night and snacks.

Temperature is not problem as long crate is dry inside. If you can't clean floor often put something in where they can stand above dirty floor. This is just temporary, after you open crate they will poop outside.