I believe now that I can completely characterize a relationship between the quantities m (number to be factored), s (least perfect square > m), r (the difference between s2 and m) and a reduced set of candidate values for c1, which will also give c2, n, p, and q.
The details are in my Google Document on this study, to be refined by further examination, at which time I will web-publish a final paper of the results, the good Lord (and anyone else validly concerned) willing and the creek don’t rise.
Ladies and Gentlemen and All Shades Between the Two, we are approaching the finish line of what I’d set out to do. What a feeling!
EDITED TO ADD: I can see, rather embarrassingly plainly now, that I haven’t reduced the set of c1 values at all. However, I am not ruling out examination of the behavior of the numbers to show me quicker ways to get from m, s, and r to c1 and n. I do not feel my work has been in vain, nor is the disappointment particularly great. Still with me is a suspicion that I am inches away from something helpful and new.