A bit OT, but there is NO WAY for the TSA to "figure out" if you have filed the required forms. Firstly, as clearly in indicated in the instructions for the FinCen 105, in the part titled "when and where to file", you can file pretty much at any point prior to departure, including filing AFTER you have cleared the TSA checkpoint. Even if you have filed, you have no proof of filing (unless you specifically asked for it) the TSA has no right to ask for any, and the TSA has no access to any information on who has or has not filed, nor do they have the knowledge or expertise to make a determination about compliance with the law even if they did illegally gain access to such records.

Secondly, I don't see anything in the regulations that prevents you from filing by post. So if you drop your form in the mail on your way to the airport, you're 100% legal, and since that form goes to the CBP, who then passes it along to the IRS, the TSA has no way to "figure out" if you have filed, and indeed, it would be an violation of the Privacy Act if they tried to find out.