Just some thoughts. It's been 15+ years since I saw a LCIII. 🙂
The nice thing about OS 7.1 is it can simply be copied from one computer to another. I do have memories of something called "enablers" which may have been required on certain machines but I don't remember exactly for LCIII (everymac.com might be a good place to check). I don't think there was the equivalent of Firewire Target Disk mode in those earlier computers. If you can find a SCSI enclosure somewhere you could try removing the drive from the LCIII and copy to it in the enclosure attached to a different computer.
Another consideration. Sometimes older computers have other issues such as capacitors gong bad with age. I don't know how to test for that but what's your objective in getting this to work?
Some cities have Mac Users Groups (MUGS) where you might find somebody who collects old parts and could give a hand. I'm not a member of one but I still have my cherished G3 mothballed in the basement. It could handle everything from low density floppy to USB and Firewire, SCSI to IDE/ATA, ethernet to Appletalk phonelink. *sigh*