The easiest solution seems to be to do it in Frosted Ultra Detail, which has a minimum wall thickness of 0.3 mm.
The firing pan should print with that and the flintlock lever still would look reasonable if you increase it to that thickness.
It would be more expensive, but the difference should not be that bad for a one-off, and you could file it down a bit.
A "supported" part is a wire or flat piece that is supported at both ends. See the detailed guidelines for examples:
https://www.shapeways.com/materials/frosted-detail-design-gui delines . I don't think there is much you could do with your design.
Tips and edges have to follow those guidelines, too, but as long as the edge/tip is not at a sharp angle and the part as such is thicker than the minimum, you are OK. I have gotten away with 45 degree edges. 90 degree certainly is safe.
Greetings
Dragoman