I’m in the same boat as you, except I already have my truck - 2019 F150 with the Fx4 package (“off road”) and the 5.0L V8. Crew cab and 5.5’ bed.
GVWR is 7000# and my truck weighs around 5300-5400 with a full tank of gas. So the payload is 1600-1700#. Sticker on the door says 1621#.
I wish I had gotten an F250 instread. I’m looking at truck bed campers and will probably purchase a Camp-X, Hallmark Milner or FWC Raven. No matter which one I go with, the camper dry will be around 1300#. I’ve done the same math as you, by the time you add water, propane, food, gear, girlfriend and dog, you end up somewhere between 2000 and 2500 which would put the GVW at 7400-7900#. Not ideal at all.
I’m also going to remove the tailgate and the back seats. So maybe that will drop 150#. Also considering dropping the gf but that’s a lot more complicated. No matter what though, I’ll be well overweight.
I’m going to get E rated tires and airbags. I talked to several people at camp x, FWC, and Hallmark and they all say my truck will handle the weight just fine. There are also many people on this forum who are doing the same thing (pop up camper in a half ton) and none of them have reported any issues. I realize that the manufacturers are biased and want to sell campers, but they also don’t want lawsuits and they have a ton of experience with this stuff so if it were a big problem I’m sure they’d let me know. And I have yet to hear any issues from anyone on this forum doing the same thing.
Anyway, that’s where I am now. Just letting you know what I’ve learned so far. If I were in your shoes I’d probably just buy a 3/4 ton truck. I might see how it rides on my truck and upgrade if it doesn’t feel safe or rides terrible. Or maybe I’ll get a small, off road worthy trailer.