I think what will end up happening this way is that whatever you throw in first will be the main contributor to your culture. By the time you throw in your next dregs, the initial organisms will outnumber it probably around a million-billion to one in terms of cell counts and as the nutrients become depleted the new incoming dregs will grow out less and less. They'll be there, but their contributions will not be as big as that of the first and second dregs. Since you desire a house culture that is a mix of these dregs, I think it stands to reason that you'd wish them all to contribute roughly equally, yes? If that is the case what I'd do is split that wort into 1/2-1 gal portions and pitch dregs into each individually. That way you'll have a number of equally happily growing populations that you can later crash, decant, and mix the slurries as you desire in order to pitch into your beer of choice to be the first "house sour."
Just my thoughts...