I wouldn't pay to pull the tank on radiator to inspect unless it was something unique/hard to find/rare/expensive. Like you said it will probably cost more than just replacing. Just my opinion.
There is probably someone smarter than me that has a way to figure what kind of exchanger is in the radiator tank without disassembling it. A small mirror in a radiator hose connection? Borescope? Drain the trans loop in the radiator and titrate fluid in to measure volume? Not sure.
Hopefully there is some type of tag or identifying label somewhere on the new radiator you can google and get some info. If it comes back as being a low end radiator than I would suspect it possibly at least contributing to your problem. The fact your engine coolant is 190ish reinforces the fact your cooling is otherwise OK. GMT800 3/4T Chevy's with a tow package have an insane amount of cooling capacity compared to older trucks. A lot has to go wrong to have an issue.
If you are somehow able to verify the radiator may be part of the issue, you have a two possible solutions. 1) Replace the radiator with OEM. 2) Significantly upgrade the trans cooler and route it first, before the small trans loop.
If it were me I would replace the rad. On RockAuto.com the cheapest radiator for your truck is $67, the most expensive (AC Delco) is $178. You have a very common platform and therefore quality parts are inexpensive, take advantage of that. For the Delco part look at the RPO codes to make sure it is the correct one. I'm not a rich guy but for $178 I would do it to rule it out.
Even if you go to the dealer and it is $500, that's less than one payment on any 3/4 ton anything these days. Keep in mind Chevy spent a gazillion dollars designing your truck so a ditsy soccer mom could hook up her 10k pound horse trailer and pull it wherever without caring at all about shift points or coasting or lockup or anything.
Again all just my opinion
