The trailer has a max weight capacity rating itself, which is likely well within the range of "C" rated tires. Going up isn't going to really do much for you except cost money.
Peace of mind. Which is worth a lot, for this chronic worrier...lol.
Yes, the stock tires are 'legal' for the trailer (which has a GVWR of about 2500 lb more than empty - not that I ever see myself carrying 2500lb of 'extra gear' in the trailer.) With a higher load range come more plies, on average. If I can find US-made tires, that's more peace of mind (although it appears that even one of the two GoodYear TT offerings now has a recall initiated because of - sudden failures (blowouts).
I have one of those HF laser temp guns, and check the temp of the tires when I've been running them hard. So far, so good - they have been showing roughly same temp as the Firestones I have on the Envoy. Yes, I know I'm exposing myself to potential criticism / ridicule by evaluating CCC manufactured tires I consider to be inferior, with CCC-mfg'd instruments to evaluate said inferiority.
Flame away!
One thing I learned a long time ago is to have trailer wheels/ tires balanced for the longevity of the tires and bearings. You would not believe the amount if weights it sometimes needs to balance them. A couple if times had to use truck weights on 8" rims! At that size, you need to find a shop with a bubble balancer or get your own.
Noted! I really need to find a RV-specific shop around here to check on things like this, along with alignment, and other things I "don't know about" re: trailer maintenance.
During my Montana trip, there were times while I was feeding the Voy yet another tankful of gas when I'd look at the trailer axles on the driver's side and swear that it looked like the rear tire was canted on an angle, relative to the front one. I need to check that, and soon. Really wish my HOA didn't have restrictions regarding trailers. Or that I was better at making up to-do lists...
The things I put up with (& the money I spend) to shorten my bucket list...LOL But I'm still glad I'm fortunate enough to be able to shorten it. Beats a life of quiet desperation, to turn a phrase.
On edit (and to nudge this thread back on track): I read yesterday from a supposed GM employee that the 1500HD was created largely as a 'badge job' so that GM could boast the largest tow capacity-enabled 1/2 ton on the market, back in the day. Supposedly, it shares much more with the 2500 than the driveline (although the same person said that the rear axle is rated for 900 fewer lbs, and is a semi-floater instead of a full.) So if I ever want to swap to the 4.10, best to get it from a 'real' 2500. That swap probably will happen at some point. (damn you, Moose! LMAO)