For some reason I figured you would know and be the one to respond. I checked out the site and though it is a LC event I did not see anything that prohibited NON -LC vehicles. With so many Toyo 4X4 I am surprised there isn't one.
Toyota did help sponsor the TTORA Rubicon run last year so I believe Toyota is slowly warming up to the offroaders.
Hopefully over time, it'll happen.
If you do plan to do the Moab run w/ TLCA, there will be 1 4th gen there to do the moderate runs. He does not frequent this board but if you are interested, I can get a hold of him and both of you can talk about the proposed trip.