Seems like a learning opportunity here. Could this have been handled differently? Maybe something like:

BIGCO asks for info/demo, you oblige. BIGCO is pleased, asks for more discussions, Q&A, etc. At this point, you present an NDA/Non-compete.

Certainly they could say "no", at which point the outcome is the same as it was here. Or they say "yes", at which point they have a little more skin in the game, and you have the law on your side.

I don't know if this is the answer, but it seems like a good discussion.