F250
I doubt that a 27 foot trailer is as light as you suggest when you actually weigh it yourself on the scales .
"I need to tow a 27ft travel trailer empty weight 4100 with gvwr at 6200. "
Until you weigh it you do not know.
But a representative sample of 27 footers would suggest that you will be 5600 # plus , more like 6200# .
i am 900# tongue , 5400# axles , actual for a 21T komfort .
Heavy tongue weight eliminates sway .
With an F 150 the tongue weight plus gear you will be at the max gvw of the pickup .