These excellent suggestions and insightful questions seem to be begging for a theological or philosophical treatise on poverty and medicine. However, I am neither a theologian nor a philosopher, but still feel inclined to offer a few points.
1. What's so cool about DPC is that any of these suggestions, and I'm sure, plenty more, can be adapted, implemented and fine tuned without worrying about insurance, Medicaid or Medicare regulations.
2. It warms my heart to hear both DPC providers and administrators have concern for indigent or financially struggling patients.
3. If your DPC is a 501(c)3 already, you're off to the races. I suspect that not many, or any, are though.
4. You could not "run" your non-for-profit scholarship organization and pass the funds on to yourself (DPC practice) me thinks. Better, maybe to consider approaching another charity or religious institution to run the scholarship fund, if you are looking to help donors with a tax right off.
5. I think that it might be worth considering a 1/3,1/3, 1/3 approach. Have a conversation with the (potential) patient and let them know that you are willing to pay for 1/3 of the membership cost. They, however, must also contribute 1/3 of the cost and also find another individual (preferably family member or friend) to cover the last 1/3. I think this adds credence to their dire situation and also makes them responsible to solicit assistance from someone besides you. (I am usually a sucker for a sob story, so having some definitive verification by a close contact of the patient is helpful.)
6. Poverty should hopefully be a transient situation (especially for the able-bodied) and so putting a time limit on the assistance program seems a good idea as Jason suggested.
Anyway, just a few quick thoughts that popped into my mind when scrolling through this thread.