Tow vehicle PAYLOAD is the key number you should be looking at for the Tundra.
It can safely pull 10,000 lbs... But how much weight can it hold? That's the trucks payload.
My 287bhsw tongue weight loaded is around 750lbs-850lbs.
The trailer tongue weight counts as payload on your truck.
In most vehicles your own weight, passengers, luggage, anything you have inside of the cab and in the bed is considered payload.
I agree the jayflight line is solid choice for a well built family trailer.
The "lightweight" lines are built with the weight of the trailer being the driving factor of the materials and methods of construction used.
Plenty of people pull similar trailers to mine, with similar weights with similar 1/2 ton trucks as yours successfully.
You will be close to your trucks limits but it can be done.
If it were me I would forget about the outdoor kitchen (you will save a bunch of $ with the jayflight), get the other things on your list. Pull it with the tundra but consider a 3/4 ton truck sometime in the future
2015 Jayflight SLX 287BHSW
2002 F-250 Extended Cab 4x4 6.8
Teconsha Prodigy P3 Brake Controller
Reese SC Weight Distrubition with sway control (12,000lb)