Nice truck. I had an '07 Tundra 5.7 Crewmax 2wd. I have a Forest River Wildwood 28DBUD. 28' box and about 6800-6900 loaded. Loaded, with fuel, wife, five year old, the dog, and 60lbs of fire wood, I was 40lbs from MVWR. It handled like it too. Wind and passing trucks moved me around a fair amount. Wasn't ideal. Doable, but not ideal.
So, to answer your question, I'd focus on trailers with a dry weight around 5K. That'll keep you closer to 6K loaded.
I wouldn't bother calling Toyota. All the info you need is on the yellow sticker of the door. Just do your own math adding 500-1000lbs to the dry weight of whatever you look at.
2016 Duramax 2500HD
2016 FR Wildwood 28DBUD