Looking at the specs, you have to remember to add to the dry weight for any options added to the trailer. If you do that and add another 1,000 lbs. for gear, food, some water, etc., I think you will be around 8K for towing. The Tundra will handle it, but you will know it when towing on steeper grades or in the mountains.
My 5.7 Tundra handled our 2003 Sunline ok, and it weighed around 8 to 8.5K. I wouldn't have been too thrilled about crossing the Rocky Mountains with it! But if you take your time, use Haul Mode and keep an eye on temp gauges, you should be ok. I can tell you that when we downsized to the Minnie and reduced the weight to 6,500 lbs., you can really feel the difference in towing.
A good WD hitch will be a must! And, I would avoid too much weight in the bed, especially if towing in the mountains.