Hi,
Perhaps 60 MPH on a 4% grade while towing is a bit to much to expect from any gas towing rig. Most slow to 55 or even 45, and use second gear. Even in my motorhome, I am slowing to 45 in second gear, and I have a 17,000 pound motorhome towing a 3,000 pound car.
When parking the vehicle is moving very slowly, and it is not unheard of for the transmission to overheat a bit. When it reached 235F+ it is time to shut it off, take a 15 minute break, and see if it cools down a bit. Perhaps more than a bit?
Some with 4 wheel drive trucks will shift into 4X low range, so that the engine RPM is higher for a given road speed while backing up, and this seems to be easier on the torque converter and engine.
I think or would hope that the large transmission cooler will solve your overheating problems. Still when you try to go wide open throttle on a hill while towing, it will warm up that transmission. If you do not want to be towed back for transmission #3, then slow down!
I might suggest washing the radiator, but you just replaced it, so it should be clean. I would replace the clutch between the fan and water pump, unless you hear the fan roar while climbing those hills. The clutch should be locking up and taking the fan from a lazy 1,200 RPM's up to full engine speed while climbing a hill IF the air going through the radiator is over 165F, and the clutch is working. This will quickly cool both the radiator and transmission.
Much of the heat from the transmission goes into the radiator lower tank, and only about 25% will go out that small transmission cooler into the air. I think that a new clutch will improve the overheating situation on both the radiator and thus the transmission as well.
Yes the E-350 is a towing beast! You do not have the 'best' year of 460" engine, but have fuel injection and a fairly stout transmission. Try running higher octane gas, and it should improve the performance. The anti-ping sensor will cause the engine to ****** the timing, and enrichen the fuel mixture to stop the pinging, both will reduce the fuel economy.
I think that the same year F-350 460" engine had a few more horsepower, due to a larger radiator and the ability to pass much more air through the engine compartment. I know that diesel vans over the years had lower HP ratings do to overheating problems. Also the V10 in vans was something like 45 HP less in a van than the truck, and class A motorhomes got the 362 HP V10's due to a much larger radiator.
If you are still overheating the engine after the new fan clutch, then consider a oil cooler. The later model vans have a engine oil cooler bolted directly to where the oil filter goes on the engine block, and the oil filter bolts onto the oil cooler. Then the radiator lower hose connects to the oil cooler, and on to the block return line. You should be able to find all these parts at a junkyard, if your van did not include them from the factory.
Good luck,
Fred.