Just want to understand the load/weight rating requirements for replacement tires

[MarkEHansen]

I have a 2019 Cougar Half Ton 26RBSWE which came with Trailer King tires.

The tires are load range E, but I'm curious about the rest of the markings on the tire. Here is what I see:

ST225/75R16 117/112M

I get the first part, and I think the second part decodes as:

117 = load index single, which is 2,833
112 = load index double, which is 2,469
M = Speed rating 81 MPH

I also see on the tire the following markings:

Max Load Single 2,860 lbs AT 80 PSI Cold
Max Load Dual 2,470 lbs at 80 PSI Cold

These seem to line up.

I assume that the single/dual is based on how many wheels are on each end of each axle. My trailer has a single wheel on the end of each axle, so I would use the "Single" rating.

The load range on my current tire is E, which all I can find means 10 Ply. When shopping for new tires, do I need to make sure they are at least Load Range E as well as a load index that is large enough to support the weight of my trailer (plus a reserve)?

For example, the Carlisle Radial Trail HD in my size and load range E shows a load index of 117M (so, 2,833 lbs and 81 MPH). This seems like a good replacement tire.

Am I thinking about this correctly?