4,000 bare? You need to consider what it will weigh when loaded up with water (8#/gal), food, cooking utensils, plates, flatware, clothes, ammo, lawn chairs, picnic table, grill, firewood, propane, cleaners, coolers, tools, rugs, coffee makers, toaster, boots, and a lot of other stuff that is going to make its way into the camper.
Also consider that most campers are a like a big sail behind your vehicle and are easily affected by crosswinds and semis blowing by you as you merge into traffic. You will want/need a weight distribution hitch and sway bar so include that into your budget.
Finally, just because your truck is rated for 6000, doesn't mean it will be comfortable hauling 6000. The F150 "properly equipped" is rated to tow like 25,000# but no one in their right mind is going to regularly tow that kind of weight behind a 1/2 ton pickup. Oh, you need to figure a 4000# trailer should have 400-500# of tongue weight for safe towing. What is that #400/600 pounds going to do to your allowed payload? With the wife and kids and #400/600 hanging off the rear, you are probably already over GVWR sitting in the driveway.
That said, you have a good truck for towing. Just make sure you do your homework and don't believe anything the salesman tells you without verifying it. They want to make a sale, they will happily sell you a #6000 dry weight camper for your #6000 rated truck knowing darn well that you won't be safe towing it.
