I did not see what year the LX is, but it is probably much like mine if it is a 1997 or earlier. Later ones need to be checked to be sure the Low-Neutral-High transfer case lever actually moves gears in the transfer case, and not some esoteric electronic gizmo. All bets are off if it does not have the lever-gear connection.
I tow a 1993 Land Cruiser, four down. Toyota all but hung up on me when I asked them, several Land Cruiser mechanics said they would do it, but not suggest I do in case of a problem, etc, etc. Somehow I was reminded of Chicken Little.
Since I had the Cruiser, with about 225K miles, we are old friends, I could not see any reason why I should not tow it. So, I decided to try it, and if I tore it up then I would get something that would work. So far I have towed it from Fort Worth, to Phoenix, to Amarillo, to Oregon, to Phoenix, to Dallas and back to Amarillo, and it does fine. Put the transfer case in neutral (solid old transfer case lever that actually moves gears, not electronics), turn the key on to unlock the steering wheel lock, then off and leave the key in the ignition lock to keep the steering wheel lock unlocked, install a Brake Buddy (or appropriate choice) to handle braking with a 5,600 pound rig, and you are good to go.
Unconventional things I have done, besides towing four down in the first place: I do not use a base plate, but attach a heavy steel Blue Ox tow bar directly to the tow hooks on the front of my truck. I saw a video where a Cruiser, full of water, was lifted completely out of the water by ONE of those, so they should be good for towing. Saved about $180+ for an ugly baseplate. I also ran a 12 volt power source for the Brake Buddy from the coach house batteries - no danger of running the toad battery down and losing braking.
I use synthetic oils in the drive train, and that may help keep it cool. The transfer case is always cool to the touch when I check it. I do start the engine and run the transmission through the gears at least once, and mostly twice a day when towing. Can't hurt, though I doubt I need it to keep everything lubed.
Christo Slee (Toyota guru extraordinary) has developed a lube pump that addresses these concerns, but I see no need for it in my rig. Remco is the recognized leader in lube pumps, etc, if you want to go that way.
That is my story, and I'm sticking to it. ;*)