I too, have a 2019 Tundra (TRD Sport 4x4) and a Hawk (shell), and it is a much tighter fit than you are describing, but I suppose it depends on where you measure!
There is indeed, plenty of space between the camper and the truck bed behind each of the turnbuckle doors, but we use that for dirty storage:
1. an extension cord
2. 1 lb propane bottles we use for the stove sometimes; they're wrapped in a canvas back to avoid unnecessary metal on bed-liner
3. um, pee-bottle and poop container jar...sorry, but yeah, everybody poops and when we don't do it outside, we at least store it in a sealed container (and homemade wag bags!) until proper disposal).
Thanks to the shape of the camper, none of these things can slide out. In very dirty environments, these areas get dusty, but not as much as I might have expected. None is big enough to get in the way of regular turn-buckle checks.
Lastly, I'd be interested to hear about others experience with the camper *moving* as the OP expressed concerned. We were on very rough roads for miles and miles and while I regularly check the turnbuckles, they don't need much tightening. When I took the camper off in November I did see that the front rail had worn the LineX bed-liner, and this isn't great, but it sure isn't moving much either way!
I like the horsemat idea, but they're remarkably heavy! Weight is technically, quite limited on the Tundra (even if nobody seems to care about being a bit over GVWR...)
