I'm not an accoustical engineer but I'm always willing to offer my conjectures as fact;-) so here goes:

Megaphones distort the sound as they amplify it (I guess you can't get something for nothing) and also direct it in a fairly narrow beam so they might do as much harm as good in that respect.

The problem of accoustically amplifying a guitar was a factor in the development of the resonator guitars (National and Dobro) in the late 20's, also resonator mandolins were produced. Again, the sound was distorted or shall we say a new unique sound was produced. But the resonator guitars are remarkably louder than wood ones.

Murray's parabolic shell idea seems great. I bet one could be made up geodesically (remember the geodesic domes of the '70s?) with lightweight rigid foam insulation panels on wooden frames that could be packed flat, then bolted or clamped together on site (heck, you could probably even assemble them with hooks and eyescrews). It's worth exploring.

My real advice would be to compromise your principles, get a nice little PA and some good instrument mikes, and save the unamplified sessions for house concerts or other quiet venues.