There are a number of points I don't see mentioned in the thread.
To start with, you didn't provide the complete tire size as molded on your original tires. A complete size might be "LT215/85R16 115R Load Range E"
I have seen no mention of your actual individual axle by axle, side to side loads. While most TT have at least a 5% variation, I have seen a few at more than 18% load variation.
I do not consider an ST235/80R16 LR-E the same as an LT235/80R16 LR-E as they are in fact different "Type" tires and are listed on different sections of Industry Standard tables. Goodyear makes both Load ranges in their Marathon line.
Info on Load Range can be found
HERE.
Here is what Goodyear says "
Trailer tires are considered a special category in the current global design standards of Tire and Rim. The major difference is that trailer tires are required to carry 10% more load than an equivalent P-metric size tire. LT type tires are actually on a lower formula and would require a significant size or load change to meet the trailer load requirements."
Also ST type tires are rated for a Max operating speed of 65 mph. Exceeding that speed rating can result in tire failure.
A switch of Load Range while staying with the same size will only give you more load capacity if you increase the inflation as it is the air that carries the load, not the tire. Info
HERE.
Changing wheels to those that can carry the increased load and inflation to match the new tires is a good plan but what are you going to do about your axles and springs? They also have maximum ratings.
You can find links to most of the tire company web pages
HERE
You didn't mention why you were "upgrading" from Load Range D to Load Range E. Was it to give yourself a larger safety factor over the max load capability of your original tires? i.e. You have weighed your TT and know tha actual individual tire loads. You are not exceeding the GVWR or GAWR but due to side to side variation have less than 1% safety factor for one position. A switch to the LR-E would allow you to increase the inflation by the suggested 5 to 10 psi while not exceeding the GAWR.
With more facts we can provide a more informed response.