Is iTunes 12.4.1 available for 32-bit Windows? I couldn't find it on the Apple website, I was only offered the 64-bit version. Maybe that's because I'm using a 64-bit client for the download?
If you make the iTunes download believe you are 32-bit Windows, it will provide you with the 32-bit installer.

Change the user agent string following instructions for your browser (this guide seems to work) to something like Mozilla/5.0 (Windows NT 6.3; Win32, x86; Trident/7.0; rv:11.0) like Gecko so it appears you are running 32-bit Windows. Visit the download page again: it should now give you iTunesSetup.exe instead of iTunes64Setup.exe.