You will be able to find many "XL" rated tires in your size. I have that very size on my truck and tow a trailer similar to yours. I towed my trailer with "P" rated tires to the East Coast-and then bought new tires (Nitto CrossTex w/XL rating) and towed it back to Utah. The XLs proved to be a satisfactory replacement and did not rough up the ride.
General Tire makes both an all terrain and street tire in that rating among other brands.
BTW-Most P rated tires at max inflation don't squirm under load. If they did I would have never made it to the East Coast without issues. No blowouts, flats, sway, nothing.