Forget the effectively part. This might be illegal and negligent if a problem occurs and you will be 100% liable, no question. I read a story that someone's own insurance didn't even cover the accident because they had an illegal condition. The person had to foot the whole thing. I don't know the outcome, but losing everything I had to something this bad is not an option for us.
Most 37 foot 5ers average 13,000 to 15,000 lbs towing weight. Depending on how your Toyota Tundra is equipped (available with the largest engine, highest differential rating, heaviest spring set, tow equipment, etc.) this trailer will still exceed the tow ratings even if you carried nothing in the truck and nothing in the trailer. But you'll probably have passengers, gear, the fifth wheel hitch hardware AND the pin weight of the trailer (15% to 25% of the GTW-gross trailer weight), plus all the gear in the trailer, too. Then you do want some margin because wind and road conditions can cause that weight to be heavier as you roll down the road.
I have a 37+ foot 5er. I towed it with a 3/4 ton Ford with a big engine and matching transmission. The weight on the rear axle exceeded the Rear Axle Weight Rating AND the Gross Vehicle Weight Rating. I added air bags but found out they DO NOT improve the weight rating. Fortunately my truck could only be improved by adding the F-350 leaf spring pack and larger tires and wheels to support the new weight ratings and I was able to solve my problem. I was very lucky. Most times it requires a new tow vehicle.
I would NOT tow that trailer with a Tundra or any other 1/2 ton like a Titan, F-150, Dodge, Chevy or GMC 1500. Even smaller fifth wheels might require a 3/4 ton. You might find a small enough 5th wheel for a 1/2 ton but you also wanted effective and I would not call a truck at its max weight rating effective. If this were true I still be towing my old 27' travel trailer with my big Ford Expedition. That was a terrible tow, easily within limits even with filled with gear and passengers, but that gas engine was screaming and struggling on any hill.