Dear Pons family,
You have produced a very interesting essay. I accept completely your rejection of zero-dimensional "point" particles, the mathematical convenience that allowed physics to replace complex bodies with a point at their center-of-gravity. The problem, as you point out, comes when 'small' particles are considered to be structureless points.
Your imaginative design addresses some of the problems and your list of "high fitness" explanations (that you list in your appendix) is quite extensive.
As you point out, most of us do not even think in terms of a "nonlocal hidden variable" (NLHV) approach and you are to be congratulated for exploring this route.
Although I agree with you that "there is no obvious impediment to a mathematical formulation" of your design, this is apparently a task that still has to be accomplished. If, while working on that task, you wish to take a break and read about another approach based on non-point particles that also attempts to address the same problems that you do, I invite you to read my essay, The Nature of the Wave Function. Due to the 9 page constraint, I do not focus on the particle itself, but on the wave (function) it induces. While it may at first appear to be the classical de Broglie particle, I explain why it is not.
Finally, congratulations on making your efforts a family project. Most families appreciate the difficulty in pulling something like this off. Good luck in the contest.
Edwin Eugene Klingman