I too built a flatbed that sits fairly low. I used 2" square tube parallel to the frame rails, mounted with tabs to the factory box mounting points, and 2" tubing side to side above that to support the 1/8" deck. It's about 2" higher than the factory bed floor, and I can see some evidence that the tires have rubbed it on both sides. (2500 Silverado, 265 tires, no lift) The flatbed has been converted to a dump bed with drop sides, and that truck routinely gets
grossly overloaded, and then run over rough terrain.
I don't think it would have rubbed at all under normal driving, and with 4000lbs of OSB on the bed, there was still about 1.5" of clearance from the tire to the underside of the bed, so rubbing must be occurring when it articulates.
If rubbing is a concern, it can be mitigated with a bit of lift and tire selection, or truck selection. (Ford and Dodge trucks sit higher, so less likely to get rubbing a lower flatbed, IMO.)
I personally would prefer to have useful space under the sides of my camper. My flatbed does have ~14" tall drop sides, and even just having those allows me to store outdoors stuff under there. It's not secure, or dry, but it still helps.
FWIW, I know the local flatbed installer orders all his beds to fit Ford trucks at ~38" wide IIRC. He did say that he would order me one at no extra cost that would fit my GMC, but that bed mfg still uses a 3 or 4" rail along the frame, with 2" channel above, so it's already at least an inch higher than it needs to be in my experience. I probably wouldn't worry about an inch, but if the bed is a LOT higher, it might be an issue.
I would agree that if you want a dedicated camper hauler, it would be neat to have a custom bed made to mount the camper lower, though you may then need a custom camper to clear the cab... I know campers for F250's leave a LOT of space above the cab when mounted on dodges and Chevy's... so that might be an option...
Another option is a utility bed, like shown in the photos above. A bit more expensive than a flatbed with boxes, but probably not a lot... Most mfg's offer a low side height option that works with some truck campers. The big issue is finding one that will allow the camper to be tailgate width behind the wheel wells. OR you have to order a camper that's skinny all the way back. I dont' know where you are, but Largo Tank and Equipment in Farmington NM can build you a flatbed with built in boxes that would work and look great.