Hey, great trailer! We towed with a 2003 F-150 for 2 seasons before wanting a 3-row vehicle.
Since you are looking used, get something new enough with a 6-spd transmission. If your 28' BH is the same 28BRS as my North Trail, then many half tons can be equipped to tow that. Payload may be a problem. Some crew cab half tons have payloads as low as 800 pounds, but most are in the 1300-1500 pound range. Take off 800 for trailer TW, 200 pounds for misc camping gear in the bed, and make sure you have enough for the family. That said, I do recommend a 3/4 ton. Ford's V10 and GM's 6.0L are very reliable, lots of power, and they all come with the required equipment for RV towing. With the half tons it can be tricky finding the ones with the better gearing and trans cooler.
If you are open to passenger vans, they cost about half of a similar age/condition/mileage pickup or SUV.