Since the van is up and running I'd try that first. Given it is a 1/2 t van, trailer brakes are a must. You will probably have to tow in 3 rd gear ( assuming a 700r4 4 speed auto ) unless it has a really deep rear gear. I'd also put maybe 400 lb solidly attached weight in the back of the van as vans tend to be pretty light in the rear.

The F250 would be slightly better however I've seen 2 versions, a light and a heavy. If you have full floating axles in the rear you likely have the heavy one with bigger brakes.

My 14 ' deck flat trailer has hydraulic surge brakes so I can attach it to anything and have brakes. My 20 ft box trailer ( box size not overall ) has electric brakes. When I towed it with my 77 Chevy 3/4 T I used a old school tap into the brake line controller. My 80 Chevy single wheel 1 ton will tow the box trailer without brakes just fine though I really should add a controller for added stability. ( The 80 has 13" x 3 1/2" rear drums and huge front discs. )

To get some sense of what will tow what, look on the u haul site for truck / trailer rental. They have a configurator that is very conservative.