yes you do need a transmission cooler, but review the size of your radiator.
I had an transmission cooler installed on a gmc truck, but its radiator set in front of the cooling system radiator and cut my cooling capacity.
I did not watch the fuel gauge, I was always watching the temp gauge.
you may need a bigger radiator, if you have a trans cooler installed.
my truck sure did,
advice from a guy that has tow a lot over the years.
get the biggest, hairiest, strongest, truck you can afford with the most brake capacity, you will never be sorry no matter what trailer you buy.
going up or down a long steep hill, you cannot have too much power or too much brake capacity!!!!!!!!!!!!!!!!