These are all good ideas guys, thanks. I think there's multiple
options. I think the idea of not holding the part in a 100% fixed
position is good. I actually think that's the biggest problem. I think
if I did fix it so the part is held, but still allows to adjust
itself, that could help. Maybe if I had the tap head I could almost
just hand hold it in a V block and kind of guide it, while allowing
the tap itself to kind of adjust it slightly as it goes in.

If I just tried using the forming type of tap in the hand power drill,
do you think I'd break less by doing that than the normal gun taps?

I would ultimately love to use the combination drill/tap. That would
be a huge time saver. But that would at the least need the reversing
tap head. The Tapmatic ones are more money than my whole press is
worth. Has anybody ever had success with any that are a little less?

Also... On the combo ones, it would seem like there would be a lot of
room to screw up. The drill portion and tap portion are obviously
working at two different speeds, so even if you were doing it with a
reverse tapping head, I would assume that theoretically you could
start the tapping portion in before the drilling portion is finished.
I'd think that using something like that would work best with
something automated, where you could control things more exactly.