When WAS the last fluid and filter change??
AND, are you sure you have proper ATF level?
First and foremost, check your ATF level....I though you'd already had so I didn't mention it before, but that's always the first thing to check on these transmissions.
That truck should be able to haul that trailer without transmission issues, as long as the transmission has enough ATF in it to remain cool under load.
I pull a 4,000lb trailer with my '97 Rodeo automatic through very hill terrain in Colorado.
The thermostat and/or waterpump are not your problem; problem lies in low ATF level, dirty ATF, low voltage to the transmission, or internal transmission issues. Has nothing at all to do with your engine's temperature...unless your engine IS overheating too?? But you didn't mention that.
Keep us posted.