Driver architecture (x86 vs AMD64) needs to match the OS architecture (i.e. x86 = 32-bit, AMD64 = 64-bit). You can't change the OS (i.e. from 32-bit to 64-bit, or from Server 2003 to Server 2008, or from Standard to Enterprise) when doing a BMR (or, for that matter, when doing a full system restore over the top of an existing install).
Hardware doesn't matter in this case - all AMD64 hardware supports x86 mode. You can install either a 32-bit OS or a 64-bit OS on AMD64 hardware, and the drivers will be x86 or AMD64 to match.