Remove either conservation law and work out the equations for an elastic collision, and you'll see a new degree of freedom pop up. In classical mechanics they are logically independent, and since the space is affine, frames of reference don't help you.
Your sketch takes as a premise, something that is actually a consequence of conservation laws: the way quantities change under frame-of-reference changes.
You can get both by looking at the least action principle under frame of reference changes and using Noether's theorem, but that's not what your sketch is about.
You've got it backwards: in an elastic collision, kinetic energy is conserved by definition. Removing one conservation law makes the whole exercise physically meaningless because elastic collisions require as a given that kinetic energy will not be converted to other forms of energy. This premise guarantees that the momentum is treated as a quantity separate from the total energy of the system (which, as a reply to your first post pointed out, when expressed in relativistic units, includes not just the energy-mass equivalency but energy-momentum as well).
In reality, momentum in almost every collision will be converted into other forms of energy such as when intramolecular bonds are broken and materials are deformed in a car accident or when momentum is converted into mass in particle colliders. The elastic collision equations are only useful as a rough approximation precisely because they go out of their way to simplify the energy-momentum equivalence to conservation of kinetic energy just like Newtonian mechanics is a simpler but useful rough approximation of relativity at low energies.
You can't use elastic collisions to reason about energy-momentum equivalence because their most basic assumption is that there is no energy-momentum equivalence.
Is there some objective sense in which conservation of momentum is more fundamental than conservation of energy in a different reference frame? Since either can be derived from the other, how do you determine which is "actually a consequence" of which? (I ask out of genuine curiosity)
Well, what is fundamental isn't the point. If you imagine a world where different quantities are conserved, there would no doubt be different change-of-frame equations describing how they are. The point is you can imagine a world where energy is conserved but momentum is not. Actually it's possible to imagine physical systems, with affine frames of reference, where nothing is conserved at all.
Fundementality is a different thing, but it's more natural to assume conservation laws and take frame-of-reference equations are a consequence. You can do it the other way (I'm pretty sure, but not 100% sure because my physics math is rusty) but you're still making assumptions, and you still don't have one conservation law following from the other.
Your sketch takes as a premise, something that is actually a consequence of conservation laws: the way quantities change under frame-of-reference changes.
You can get both by looking at the least action principle under frame of reference changes and using Noether's theorem, but that's not what your sketch is about.