Dan,
A couple of things come to mind. Can you boot into safe mode. If you can, you should be able to do your copy and then you're good to go.
If you can't get into safe mode, boot your Win98 CD. Before you get to the actual install, there should be a few screens that will allow you to bail out with F3. Bailing out will take you to a screen with a prompt; you might be able to run your copy from there (DOS will be loaded at this point). Only problem I see here is that your loaded OS will not see FAT32 drives.
If neither of these work, use your other PC to download a bootdisk from:
http://www.bootdisk.comBoot the problem PC with your new bootdisk. This should give you enough of an OS to do your copy.
Good luck.