While I agree a larger than stock tranny cooler in that truck is a must, he has not exceeded what the stock tune or engine setup was designed for. He has not ever put a scanner on the thing yet. My 2002 6.0 reads 210 on the gauge in reality according to HP tuners that is anywhere from 190 to 218. The newer gauges are even worse.
The 5.3 is an LS based motor, iron block, iron heads. The difference between that motor and the one is my Trans Am is iron block VS aluminum block, 5.3 higher compression heads, updated oiling in 2006 and 5.3 vs 5.7. The difference between my 2002 6.0 is the 5.3 has higher compression heads, updated oiling and 5.3 vs 6.0. I have seen my 6.0 hit 220 to 235. This is not an issue in these engines. I do like to keep the tranny fluid under 200 and with a larger cooler I have not seen over that yet.
Scanner and pull a bit before you start spending money. Do a flush if it makes you feel better. Don't just start swapping out parts.
Based off what he says the only thing is his truck is working harder to move a larger trailer.