Okay, got a small problem. Turns out I completely forgot that I tossed the floppy drive when I got this old computer from my friend. So here's my plan...
I downloaded this "Ultimate Boot CD" ISO with all these utilities on it. So does a CD-R automatically finalize when you burn a CD? If not, I just add the ROM file to the CD, and since there's already a flash utility on that CD, I could just go right ahead and use those programs from the CD instead of a floppy. The only thing is, I can't backup my old BIOS, but I'm sure I've got the right version anyway. Also, if I added that ROM file after, that means it'd be adding another session. Can DOS read all sessions on a multi-session boot disk?