I would name "intrinsic" those properties as mass, charge, spin, which are frame invariant and "not intrinsic" what is not frame invariant, but it's terminology. Don't know the meaning you attribute to the word "ontology" here: momentum is a (very real) physical quantity.Nor the same frame will agree with its own previous value just by changing units.
Those "values" are not beable/ontologies in any sense, their usefulness is not to be denied, but they are just labels, not true statements about what is real.
The math actually tell us something important, especially if you think symmetries are real, not artifacts of "frames".
Because what "counts" here is the momentums difference between your navel and the truck, not their absolute values. Anyway this is a different story from the fact momentum is frame dependent. In this last case momentum is important, and is conserved for an isolated system, providing the frame doesn't change. And if it changes, we simply have to remember to ricalculate all systems momentums. (This is a common mistake among laymen, as you probably know).So if I take my navel as a frame, I don't change the momentum of a truck heading straight onto me by jumping out of its way. I know it "for real" because it does not require the same change in momentum.