I wouldn't be blaming made in Ch*na tires right off the bat. There's a lot of things that can contribute to failure in ST tires before blaming where they are made. To be sure, some of those tires aren't great, but they're not all bad either. Besides a puncture, it could be anything from improper care and abuse to suspension or frame problems. And besides mileage, some say you should replace your ST tires every 5 years even if they look okay. One final note, insist on the tires that you get being fairly recently made. They often sell tires that have been sitting on a warehouse shelf somewhere for a very long time.
You'll find some interesting reading on this site: RV Tire Safety
We're on the west coast, and by the time a trailer gets delivered from Indiana, it has over 2,000 miles on the tires and they are well used before you
even use them.... You also have to know that the driver hasn't necessarily treated the tires with care either. It's not right what the RV manufacturers do.
Maxxis is an ST tire that seems to be highly recommended. Many also recommend the Goodyear Marathon but some have not had good experience with them. But again, it may not have anything to do with the tire itself. If you go to RV.net, there's lots and lots of discussions on what the best choice is. Be prepared to get a bad headache though....
If it were me, I'd be replacing the whole set. Maybe not completely necessary, but I'd want to minimize the chances of getting another blowout again. I'd do some reading on what needs to be done to take the best care of ST tires and then decide if you think you've done that. You might also weigh your TT at a scale and see how that compares to the load capacity of your present tires (don't forget to take the tongue weight off the axle weight). For a TT that size, if you have load range C tires, you might be better off going to D tires.