Your "E" rated ST tires are rated at 3420 lbs., more or less, while "E" rated LT tires of the same size are rated at around 3040 lbs. Which may still work. Your trailer's GVWR is 12,000 lb., and you'll have around 2400 lb. pin weight, so that leaves around 10,600 lb. on the two axles. So if your trailer has 6,000 lb. axles you would be fine, if you have 7,000 lb. axles, you could still be okay, but won't match up with your axle ratings.
If you go to a "G" rated tire, and higher pressures, you will have to ascertain whether you wheels are rated to the higher pressures you can run with a "G" rated tire. My 2010 Fuzion came with "E" rated ST tires like yours, but it also had wheels rated for 110 psi. So it's possible that yours are too.
IMO "G" rated tires for you would be unnecessary. Provided that your weights are correct.