I've run different load rating without problems, but without weighing/knowing your rig's axle weights??? How do you know you need higher rated tires on the rear? Excessive sway is caused by not enough tongue weight... I have not used WD in 25 years, especially towing with a PU that handles better with plenty of weight on the rear axle.
Spend the $10-15 at a truck stop to weigh your axles and then ask your question. Tongue weight would help too but harder to weigh. You can use a bathroom scale and a board/lever to give you a 1/3 or 1/4 ratio.
If there is not any scales close by then I would shift 200 lbs from the back of the trailer to the front and see if you still have excessive sway BUT this does not answer your question of needing better tires on the rear.
