I wouldn't give any concern to needing additional transmission cooling in your E350.
Do a search in these forums to see if it's an issue. I've never seen the topic brought up with respect to the Ford V10 engine in a Class C motorhome.
We don't tow anything except for a fishing boat occasionally with our V10 Class C motorhome, but have never had a cooling problem doing that or in any other situation. This includes traveling in Death Valley pulling long grades (haven't tried DV in August, however) and sitting in a scorching parking lot with the engine idling while the cab air conditioning is keeping us cool when we eat lunch.
In fact I consider our V10's excellent cooling to be a feather in the cap for Ford engineering. My current Chevy 4X4 pickup overheats in extreme temperatures, and the Dodge van we had years ago that we bought new with the optional cooling package on it used to overheat in extreme temperatures.
However I will add that the taller rear differential ratio in your E350 (the E450 is geared lower in it's differential) may cause slightly more transmission slippage than what would occur in an E450. As you may know, tranny slippage does raise temperatures a bit ... but there are many folks in these forums that tow things with their E350 Class C motorhomes.