I agree with other replies that the Tundra is under sized for the Designer. I had a much smaller travel trailer that I pulled with a 5.3 Silverado 1500, and it really struggled going into strong head winds or up steeper grades. And I didn't think it was a safe tow let alone the overtaxing of the pickup's drive train. The tail was definitely wagging the dog!
Yes, a 2500 diesel would be a much better match. However you will still want to make sure that the 2500 is capable of handling the pin and towing weights of the 5ver. Or consider a diesel 3500.
Good luck with your choices.
'06 2500 Silverado 4X4, Duramax LBZ Diesel, Firestone air bags
'08 Tracker Pro Guide V-16 Boat
'12 Pinnacle 31RLTS