Outside shower faucet freeze-up...

[Gaffer]

Would you be able to use a block of styrofoam directly behind that faucet, like a 1" or so thick piece shaped around the 2 hoses to protect the backside of the assembly?

Try Styrofoam BBs like in a bean bag chair in a large zip lock bag . Will conform to odd shapes. When not quite full and the air is forced out.