Instead of making what if comments about your question we need more information to go on. Not what the books says but what your truck says.
What is the payload sticker on the drivers door that says what your max payload capacity is?
Which Tundra is your truck? There are options that will affect your payload numbers greatly.
What are the numbers on the trailer? Is the 7800 dry weight or GVW? That makes a huge difference.
Dont be overly concerned with the GVWR of 10,000 pounds. The GCWR of the trailer/truck and the payload weight are ones that should be the starting point of figuring out. Remember payload weight is all the people, pets, cargo, gas and the hitch weight totaled.
These are the figures most of need to give honest opinions and then you will have to decide for yourself which way you want to go.
We were just in the market to replace a 2001 F150 that we towed a 7800 gross weight trailer with. It towed it but was not always pleasant. We looked at the new F150 EB as well as the Tundra. The Tundra just didn't have the payload of the F150 EB with max tow package. But even that was pushing it so we bought a 2013 F250.