I also had problems trying to install 7.1 on my LCIII, even with the Enabler. I had to take some convoluted steps to get it to work. Simply copying Enabler 003 to the 7.1 install disks won't work, since it doesn't know to install the Enabler with the rest of the software. Here's how I got it to work:

1. Copy System Enabler 003 to a System 7.1 Boot Disk System Folder

2. Boot off of the Boot Disk

3. Once in the Finder, copy the Boot Disk System Folder to the internal HD of the LCIII (I formatted the HD before copying)

4. Reboot from the internal HD

5. Run the System Software 7.1 installer disks as normal

That's it! The installer won't delete or overwrite the Enabler, and it will install the full 7.1 Software into the System Folder with the Enabler in it.

Hope this helps