Welcome to the Stovebolt Paul!
You've come to the right place for lots of information and a few opinions on these old trucks.
I own a '51 1-1/2 ton and a '52 2-ton, so I've a pretty good idea what your new truck looks like.
That 2-ton weighs more than 5,000 pounds empty and depending on wheelbase and hoist might be over 6,000 pounds.
The dual wheels on the back of your truck are 20 inches in diameter and the tires stand about three-feet tall. The wheels have ten holes. The fronts are the same, but only have every-other stud, meaning only five nuts hold them on.
The car type 5-speed and rear axle you describe are not rated to carry the load of just the empty truck. Will it work to drive it to Florida? Maybe, but I would not bet on it. Will it work carrying a load like your trike and trailer? I'd bet against it.
This truck might make 50 mph as is, but could top out at 45. Rather than install a T-5 I'd try to find a Brownie (an auxillery transmission) that will allow the truck to run closer to your desired speed with the current motor, transmission and rear axle.
I would not swap motors unless the one you have now doesn't run. The 250 is not a direct swap into the '51. You'd need to deal with bellhousing and motor mount issues in addition to wiring and linkage stuff.
I think if it were me I'd fly to Nebraska and try to drive the truck to Florida as is. After checking the brakes real carefully...
