If there is a Floppy based version of the BIOS update, use that (if you have to, you may need to buy a USB based floppy drive or barrow a friend's).
Other ideas include creating a BartPE/UltimateBoot CD for Windows (UBCD4WIN)/Windows PE CD and trying to install from that.
Is there a specific reason you are updating the BIOS? When it comes to BIOS, I GENERALLY follow the thinking that "if it ain't broke, don't fix it" - if you need some new functionality of the new BIOS, that's one thing, but if you're upgrading just because there's an upgrade, at least with BIOS, I don't consider that wise.
EDIT: Also remembered - there are boot CDs you can make - DOS boot CDs that can skip the floppy - I've only done it once and it worked ok. Check out www.bootdisk.com (but just remember, you need a DOS version of the BIOS updater for that to work)
And just to clarify - 64 bit versions of Windows have ZERO support for 16 bit applications, so that 16 bit app will never run. If there's a 32 bit version, that MIGHT work, but apps that interact directly with hardware like that often NEED to be 64 bit on a 64 bit system. (Although I suppose if a 16 bit BIOS update app works on a 32 bit system, then a 32 bit BIOS update app might just work fine on a 64 bit system)