Any Mac from certain models of the SE all the way up to the beige G3's will do what you want. For what you want to do, I'd say the cheapest way to do this would be to get an old SE, the two-floppy version, with one floppy being an 800K drive and the other being a SuperDrive (my parents have such a machine). If you want a Mac that can be used for more than floppy copying and playing vintage games, though, you might want to go with a beige G3, equip it with a USB card, and connect the Imation drive to it. Using either of these two methods, you can stick the 800K disk in one drive, the 1.4M disk in the other, and simply copy files.
You can, of course, do this on any Mac that has only one built-in floppy drive (as long as it's a 1.4M drive), but you will have to either do a lot of floppy-swapping or do the copy-files-to-the-hard-disk-then-to-the-other-floppy routine.
Hope this helps to some extent.