Honestly, based on the entry / exit method, I would not mount anything either on the seat back, or the truck floor. That dog is making an upward leap that’s probably 85-90% effort, not much margin for error. Rifle sticking out from the seat, if he ran into it on the way into the truck, big vet bills. Even covered, you’re taking away probably 5 - 6 inches of his landing pad for the length of the rifle.
A floor mount, even a partial floor mount would make for an uneven landing surface and that just looks like potential knee/cruciate & foot injuries.
On a stock Raptor there’s some available behind the seat storage, not much, but it’s there. You could fit a carbine behind the seat in some orientations, but you’ll need to make sure the seats can be raised with that stuff behind the seat.