Actually, WinME still has a system disk utility, they just buried it.
I'm doing this from memory, so I may not get the exact words correct, but:
1. Open Control Panel
2. Open System
3. On one of the four tabs, I think it's the last one, is an option to make a Restore or Rescue disk (something like that).
4. Create one.
It adds a couple of utilities, but you should still be able to add your flash utility, your new bios, and a backup of your old bios.