Well, it should work with the Daystar PowerCache and Turbo040 upgrades in the LCIII.
The caveat to that is that there were a couple of different versions of the PowerCache (68030 upgrades) and apparently some of the older ones (?) don't work with all of the adapters.
I'm kind of vague on those details, but a search will probably turn up some information. It's been discussed in these fora before. There's something about the P33 (writing on the board) version and the non-P33 version.
Might be interesting to try it with a Turbo601 upgrade seeing as how the LCIII is pretty similar to a IICI and the Turbo601 is more or less made for the IIci.