Hi John...
and welcome to WTW from an east coaster!
A couple of thoughts come to mind and Wandering Sagebrush has already hit upon the most important. Size DOES matter once you start putting all your stuff, dogs and you inside the camper...especially during bad weather!
Its kind a like what a safe dealer once told me..."I've never had to replace a safe because the customer complained it was too large".
We have a Hawk. It started out on on 2003 F-250 crew cab, 7.3 diesel. GVWR is significantly less than the current models. We quickly found ourselves over placarded weight GVWR. I will not digress on that subject as it is hotly debated and plenty of threads/posts herein regarding same. Anyway, I now drive a 2016 F-350. I have all the payload capacity and towing capacity I think I will ever need. I can pull my boat with the camper in the truck bed, put my motorcycle carrier in the receiver and tote my KTM dirt bike, tow an adventure trailer or wheeled toy box, and never be worried about GVWR or GCWR.
Anyway, with 2 labs (now 1 but soon to be 2 again), 2 humans and bad weather forcing us to enjoy each others company in the Hawk makes for some very tight quarters. Especially when a lab decides she wants to stretch out!
Having said all that...our other vehicle is a Tacoma TRD Offroad crew cab long bed. So who knows what that might turn into one day. If anything, I see the Tacoma with an over the bed tent rack/Tapui tent. I like the Tacoma for places I could not or not willing to take the F-350/Hawk (ie., Alpine Loop, CO (certainly do'able in a fully loaded F-350/camper, just not by me!)). Albeit, such a Taco camping trip will be without the wife. She much prefers "glamping" in the Hawk!
So back to your question...
F-150 with a Hawk with a caveat...
If you travel light, the F-150 with fit your needs.
If you plan on adding a winch, recovery gear, and a lot of camping gear (outdoor kitchen, Partner stove, etc) you will find yourself at the very edge or beyond of the F-150 weight envelope.
Frankly, I can not see myself camping in a Fleet with two labs, even with the wife at home!
Tough decision and either way, both are great choices! And as others have noted, regardless of your choice, some suspension modifications and Load E tire upgrade will be required to level/carry the load.