Hey there, This can get very confusing, and from the looks of things your right smack in the middle of a bunch of numbers. So here it goes, Yes your Tundra has a towing capacity of 10,300 lbs. But having said that you have a number of different things to contend with as well. First being GCVW, Gross combined vehicle weight. Im guessing your truck is the 4x4, so your GCVW is 16,000lbs. That means your truck and trailer combined with every little piece of cargo, right down to that bottle of water in the cup holder, cant exceed 16,000 lbs. Now your talking about pin weight, They have a limit there to so guys dont go loading up with tidy tanks and 5th wheels etc etc staying under there 16,000 but putting to much on their rear end. Pin weight does not get added to the equation of gross combined vehicle weight rating, because your the weight of the trailer and truck makes up the GCVW but its something you need to be aware of. You need to make sure you dont exceed your rear axle rating. You can check that out, by going over the scale without trailer, all fueled up, and write down your axle weight then to be on the safe side lets say its 2000lbs and your pin weight is 1422, then just add it all on the rear to see if your under. To find out your GAWR check the sticker on the door of your truck. So heres the things you need to check.
GCVW - I think yours is 16,000 Thats truck and trailer and everything in it
GAWR - Max axle weight
Towing capacity - 10,300
From my experience.... a 30' trailer is a bit big for the Tundra, You mentioned the curb weight is 5345, That sounds REALLY low for that trailer. Heck your truck almost weighs that. I find it hard to believe that it could weigh that little.
Have you had it across a scale?