I have a 2010 4-runner 4.0 liter (V6) with tow package. This vehicle shouldn't be much different from yours in capability. There are many factors going into the tow rating, but I would suspect transmission & cooler are the limiting factors.
As for me, I recently purchased a 24ft Surveyor at 4100 lb empty/4500 lb as towed. My 2010 4-Runner managed just fine in 4th gear on level to slightly inclined hills, needing 3rd gear on steep hills. Sway control and vehicle leveling was very good with weight distribution system. Seabee - looks like you'll have ~1/2 ton heavier trailer.
For trips less than an hour, the 4-Runner seemed to manage just fine. But when I started doing 2 hour trips, something up front smelled hot - like transmission fluid. Maybe it was the brakes or transmission or engine - never really investigated too far. If I was to upgrade something, I would have started by adding a larger transmission cooler - like you did. But I took advantage of the oppportunity and told my wife, rather than risk her baby (the 4-Runner) that I should buy a truck to pull the trailer. If you think that's bordering on blackmail ... what can I say? I got a truck out of it - picked up a Tundra 5.7 liter and it pulls like a dream. Haven't regretted it since and we are absolutely loving the Surveyor/Tundra combo. Oh, and my wife thinks I'm a saint for saving her 4-Runner and making the sacrifice to drive the truck around. WIN-WIN!
Good luck in your decision.