Please keep in mind that building a "camper" is a matter of choosing what you want/need and accepting that there are consequences. For example, a pop-top camper can be easier to fit under obstacles on and off road, but, have less insulation and require set up time in snow and rain compared to a fixed-wall camper. It might fit in a parking garage, but, it has more moving parts than a fixed-wall camper.
Make a cardboard camper floor with cardboard cabinets to see how a small space will feel. Cheap, easy, and effective to try.
Make sure to order the snow-plow/camper option for the heavier duty springs and for the swaybar, especially for SRW.
An east/west cabover bed is likely easier to get in an out of for one person (and the cabover section will be shorter), but a north/south cabover bed is likely easier to get in an out of bed for a couple (at least for the person who would have been climbing over someone in an east/west bed orientation).
It would be easy to make the camper 8' wide to match the width of a DRW truck. Lots of room compared to a narrower camper, but unable to fit down the same tight trail a SRW F-350 could.
A flatbed pushes the floor higher in the air than a pickup truck bed, but can make better use of the space where the "useless sheet metal" of a pickup truck would be.
You are a welder, so, you might consider making a "non-flatbed" that has wheel wells. This would allow you to keep the camper center of gravity as low as possible. You could also build external-entrance boxes under the camper/bed.
You could make a slide-in camper with an insulated and heated basement for water tanks and batteries. Start the main floor just above the pickup bedrails. Doing so makes it easy to have lots of headroom in the cabover because the main cabin roof is so high in the air. Although the slide-in wastes some space, it does allow for easy swap to another truck if need be. (Keep in mind that pickup bed dimensions vary on manufacturers and year of make.)
You and your dog might be able to make use of a pass-through by removing the rear window and sealing your camper to your cab with a bellows of some kind.
A 2021 F-350 Crew Cab Long Bed (CCLB) has a 48 gallon fuel tank versus the 34 or 29 gallon tanks the other F-350 pickpups have.
A DRW will give you the most stability, allow you to build 8' wide without overhang, and have the most payload. I think a 2021 F-350 4x4 DRW CCLB, with the right options, has a payload of about 6,600 pounds.
Pickup truck frame bed mounts vary from. Manufacturer/year/model/bed-length can all change things. Chassis truck frames are designed with two straight rails to make it easier to move a bed/box from one truck to another.
Many DRW trucks have been used in snow and mud, and many still are.
For a bobbed-back idea or two, see the camper photo at:
Thanks guys The wife thinks I should make the round vent cover into a rising moon..... :ROFLMAO: But even with my OCD, no way I bothering with the vents any further. They fit the bill, and are designed with function in mind. They are both stainless, and I could have left then as they were...
expeditionportal.com
Even several of those propane tanks won't be an issue on the camper you are describing.