I managed to get this working by reinstalling Qt from scratch, and this time specifying 64-bit MSVC as the compiler, instead of 32-bit. There really isn't a 64-bit version of the compiler--there's only one installation package for Visual Studio, and it winds up in C:\Program Files (x86) on a 64-bit system, but it includes 32-bit and 64-bit debuggers. I had chosen 32-bit when originally installing Qt, because I wanted to make sure I'd get the same 32-bit executables regardless of which machine I built on.
So I can now build on the 64-bit machine, but I'm getting a 64-bit program, which isn't what I want. This isn't an enormous program that needs more than 4GB of address space, and I'd like to be able to deploy it on 32-bit machines. I guess my only solution is to settle for 64-bit builds while I'm debugging on my 64-bit machine, and then rebuild a 32-bit version on my 32-bit machine when it's time to deploy. That's irritating, but I'm going to have to build a Mac version on my Mac, and a Linux version on my Linux machine, too, so it's just more of the same.