Any chance it could be the cooler, stopped up or just need a bigger one without something internal.
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There is no "internal" trans cooler, just the one behind the bumper.
Yes, you could get a bigger one, but typically a stock truck doesn't need it, although in my opinion it's always good to increase the amount of heat ANY cooler can exchange.
I missed how much you're towing? If you're towing less than 12-14k, I would have to say it's not the cooler
Would it be beneficial to blow out the lines and try to clean out the inside of the cooler first
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Sure, can't hurt as long as you remove the lines coming off the trans.
But again, I would say it's not an in-efficiency of the cooler that's causing the heat to climb/spike, and "if" there's a build up in the lines or cooler itself I'd imagine that what's building up is clutch or friction material, or both, and that would be indicative of a bigger issue.
Your best bet is to try and see the data from the trans as your driving, live time, and watch what it's doing.
Things to consider:
If you're towing anything reasonably heavy, especially in hilly terrain, regardless of land speed, you should be in TOW/HAUL mode, period.
Not doing so puts added strain on the clutch pack that's engaged and lessens the trans life and also typically adds heat. It's a diesel, constant RPM is fine, even if it's higher RPM but you're maintaining a gear, that's much better for the trans and typically the engine than lugging it or trying to keep the RPM lower and possibly creating more shifts.
Fluid changes and spin on filter changes are paramount to trans longevity, especially with a truck that works for it's miles, what has the maintenance schedule been like for this truck?
You mentioned that the cooling stack is clean, that's great! Fluid to air coolers have to pass air to make an efficient heat exchange, is the core totally clean all the way through or just visibly clean? You'd be surprised at how much and what you'll find in between the fins. Pressure washing it when you spray the truck off to wash doesn't get it totally clean and you're really only pushing debris into it, not out.