The CMPXCHG16b requirement only applies to the 64-bit installation of Windows 8.1; the 32-bit version should work just fine on your machine. (I say "should" because I haven't been able to try it.) Now for the bad news: There's no way to use the free Windows 8.1 update to downgrade from 64-bit to 32-bit, even if you try workarounds that skip the Windows Store and let you burn the update to a flash drive. You'd have to buy the 32-bit OEM version of Windows, which starts at $100, and do a fresh install. (Again: That's in theory. I have not been able to confirm this.)
Or you could call Microsoft support and complain loudly enough to try to get a free copy. Because to be left marooned by shifting hardware requirements in a "point" update, with no way to downgrade to a compatible version and the threat of discontinued support lingering, is absolutely ludicrous.
I'll let this irritated forum-goer have the last word: "It's just amazing how Microsoft chooses to leave some of its users hanging out to dry because of these CPU related requirements."