Is it possible to switch parts between Verizon and AT&T iPhone 5?
You can use the battery from the phone that is not from your provider, but not switch radios. Batteries between models, e.g., iPhone 5, are the same even when the radios are not. Even when radio types are the same, e.g., GSM vs GSM, the operating frequencies, etc., often aren't.

Other pieces, such as screen glass, case, screen layers and the like are also interchangeable. Basically, it's the radio-oriented stuff that you can't swap.