Wrong Bios

You've done screwed up bad™. OK, you have two options. Your cheapest and most convenient bet (though also most unlikely) is to use a chip-burning tool to re-flash your BIOS without your mobo... but not many people have access to these. Next best bet is to tell your mobo manufacturer all about it, wait for them to stop rolling around on the floor, and send in your BIOS chip to have them re-flash it or replace it... but this is more costly and time consuming.

The problem is your BIOS holds information on how to use your mobo... including how to access your floppy drive (which is how you're likely trying to update). If it's inconsistent with your hardware, you're pretty much screwed.

That all assumes that the BIOS is completely malfunctioning. You cannot load a BIOS on a computer with a partially functional BIOS if it passes its own self-checks (as it sounds like it is doing). If you could non-destructively convince the machine that the BIOS is completely borked it may have a backup BIOS (I've seen some with a backup chip) or a failsafe that only allows floppy access (just good enough to re-flash)... though I don't know how you'd do that.

I used to have a BIOS that had the failsafe in it - somehow my BIOS was completely borked, but it gave me a prompt and access to the floppy drive (no boot disk) and I re-flashed the BIOS from there and recovered. The trick is to let the machine know it's borked.