Bottom line, you've been asking about the suitability of your old tundra for months now.
You're prepared to get a bigger truck if wanted/needed. Pretend the internet doesn't exist, because you're getting bogged down by the minutia.
Go buy the trailer you want, hook it up, tow it home. If you don't like how it pulls or then feel a newer or more capable truck is the answer, then you were already 90% aware of that, go upgrade trucks.
The world isn't going to stop spinning if your 500lbs over your payload rating or have to hit 2nd gear at 38mph to pull a particular hill. But it will demonstrate the preference or necessity for a bigger truck if the performance of your current truck is unacceptable.
A similar size trailer in length and frontal area will not pull a lot different whether it's 5klb or 6 klb.