First off, transmission ID: If you have an "OVERDRIVE OFF" button, you have a 4-speed. If you have a Tow/Haul button, you have the TorqueShift 5-speed, which is really a 6-speed but it only uses 5 gears at a time. The TorqueShift came out in the 2005 V10 chassis, but motorhome makers sometimes build on the prior year chassis.
As for the cooler.... You have an "in-tank" oil-to-water cooler in the radiator, which then feed an oil-to-air cooler in front of the A/C condenser. IF you stay within chassis specs, and you don't do unusual activities like off-road your motorhome, or back a trailer up steep, long driveways, the stock setup is usually fine.
Now... if you exceed the combined weight rating of the chassis (like I do), or go off road or back trailers up steep long hills (which I also do), then a larger oil-to-air cooler is a good investment. I run a TRU-COOL MAX 40k cooler, which replaced the stock oil-to-air cooler. This is a suitable replacement for either transmission, because it has 1/2" lines, and comes with fitting to adapt to the 4-speed. The 5-speed requires the larger lines to maintain flow. This is not a cheap cooler, but it works. Other cheaper coolers either won't have any more capacity than stock, or will not meet the flow requirements for this chassis.
A trans temp gauge is a good idea too if you run hard often. For occasional hard use however, an OBD2 reader like the ScanGauge is good enough.
Safe temp for the 4-speed is 230F, but if I hit 215F then I dial things back in case its still climbing. The 5-speed is rated higher. I've heard 250 but I think Ford actually says something like 270. It is designed to run hotter than the 4-pseed for better efficiency, and even includes a thermostat to allow it to warm up, unlike the 4-speed which can be over-cooled.
2000 Ford E450 V10 VAN! 450,000+ miles
2014 ORV really big trailer
2015 Ford Focus ST