that's a high load for a 15 inch tire, if you have room for 16's then you will have a better choice to choose from.
best 15 inch I know of are
THESE load range E 2830 lbs each
just a point to note, the tires won't see all that weight, if you subtract the tongue weight
off the gross, the remaining figure is what the tires will see.
I used to be an LT tire person, but I gave these a try and they have been good for 4 years now.
I would be happy to use them again.
LT truck tires won't have the load capacity for the same size tire, you would have to go to 16 inch.