"What do I need for a tow vehicle?" is, typically, a popcorn and beer question. It comes down to what is your personal safety margin--right at the manufacturer specs or some under. If money is limited, then that becomes a factor as well--at least for me it was. Above thread is correct--OP needs loaded trailer weights to start. I would be surprised if OP has loaded trailer up to 3395 with personal possessions--more like 2000 to 2300 which likely puts the trailer within the capability of the 3.5 EcoBoost--the only engine available that year.
But, safety also depends on proper WD hitch set up so that weight is returned to the front axle.
OP--if the issue is speed going up mountains, a gas 250/2500 truck is not likely to solve that problem. The 3.5 EcoBoost has a great torque curve for towing. It would take a diesel to beat it. Plus a family of 5 would need to get along well to load into a crew cab pickup.
My opinion--yours will vary.