Is it possible to use AT&T 3G on a Verizon LTE iPad?
Turns out I wasn't the only person that's been interested in this. There's an article on MacRumors that states that this is indeed possible, which is good news for me. The person that discovered it wrote:

Big question on everyone's mind buying the Verizon iPad was whether data from AT&amp;Ts 3G network would work on this iPad, or if Verizon/Apple would block this.
I was one of the first to obtain a Verizon iPad and can happily confirm that this is allowed! I used my ATT iPhone 4S sim card and took out the Verizon sim, and data worked! You must apply the AT&amp;T APN carrier settings before this works though.
BTW APN carrier settings can be changed here: unlockit.co.nz

edit: I just tested this and it works. Changed the APN, put my iPhone SIM in, and it worked.