Your conscience? And whatever the penalties are for signing a false statement?
I'm sure people have signed false affidavits. Here's the way I look at it: A lot of Americans travel to Cuba outside official channels. They've been doing it for years. If you're going to be one of them, assume the risk 100% yourself. Don't put a U.S. business at risk. CTS is not supposed to be taking you to or from Cuba unless you clearly fit into one of the 12 categories of approved travel. I'm assuming by all your questions that you don't qualify.
Why can't you go back the way you came (via Haiti)? You'd be dealing with a non-U.S. airline that's under no obligation to determine if you qualify for travel to Cuba on a route outside the U.S.