Just as a general FYI, the spec weight on most things is dry. The actual weight in 'as used' condition will likely be much higher.
I would say you Tacoma will be waaay overmatched, especially in Truckee. Can you do it? Probably. With a good sway control hitch and brake controller. Can you do it legally? Maybe - you'll certainly be closing in on your max tow rating. Can you do it comfortably? No, especially in the mountains.
As for the FJ80, It has the heft to handle the weight of the trailer better, but my (very limited) experience tells me they can barely get out of their own way on flat ground at sea level. I've heard plenty of owners complain about not being able to maintain highway speeds in the mountains WITHOUT a trailer. Put 4500 lbs behind it at 7000 ft pulling a grade, and let's just say I hope your dad has a thick skin as there might be some unpleasant gestures shot his way by the 1,865 cars backed up behind him on the road.
A half ton or better truck would be my preference.