The American Red Cross will not accept my blood because I lived in Europe for a few years in the early 1980s when the issue of "mad cow disease" became known. They told me that it is possible my blood is tainted with the disease and they can't test the blood for it.
I would donate blood regularly if I could and used to do so before the blood banks decided that European residence in the early 1980s was a disqualifier.
In the question at hand, however, I'm not sure that a blood drive at a traditional Catholic chapel is truly appropriate. Is there room outside the actual church for such an activity? There is not in many traditional chapels I have visited. Is the secular nature of the American Red Cross truly appropriate in a traditional Catholic chapel? Would such an event cloud the minds of the public as to the nature of the True Church? Would such an event cloud the minds of the faithful?
I don't think the answers to these questions (other than, perhaps, the first) are obviously apparent. Even if the blood drive is not through the American Red Cross but a local blood bank, I think the questions are still apropos. If such an event would scandalize even one person (even if the reason for the scandal was irrational) the so-called "greater good" would not outweigh the damage caused by the scandal. There are simply too many available and eminently appropriate outlets for a blood drive in our society to risk the soul of one person.