I don't think you'll be under 5000 lbs. The UHaul vehicle trailer weighs 2200 lbs empty, and I can't believe a Toyota pickup, even 2wd, weighs under 3000 lbs. It's probably more like 3800. Also, UHaul requires that the towing vehicle is at least 80% of the weight of the towed package (trailer plus load).
You might want to go to the UHaul website and see if they will even allow the combination of tow vehicle and towed weight that you are proposing. UHaul is fairly conservative, and generally will not allow you to get to the mfg. towing limits. For example, even though my '06 is rated for 7000 lbs towing, according to UHaul I can't rent a vehicle trailer from them and load my 4200 lb '94 4runner on it and tow it, even though the loaded trailer weight is < 7000 lbs. If I have a Suburban or full-size pickup it's OK.
You can lie to them about what you are doing, but then if something happens you are on your own liability wise.
Edit: I just checked, and a mid 80's pickup weighs 2800 lbs, so you'll be just barely over the 5K limit. UHaul might be OK with it
Check here for UHaul's rules.
https://www.uhaul.com/Trailers/Auto-Transport-Rental/AT/