I was in exactly this situation at UTMB Galveston around 2002ish. I ended up in a room with the manager who decided to implement the policy & one of the hospital's lawyers. The lawyer was patiently explaining to the incoherently-sobbing manager that:

1. UTMB Galveston was indemnified from her mistakes by her employment contract;

2. She owed me something like 60000$ in unpaid compensation, and that the State of Texas would probably penalize her another 250-500k$; and,

3. A cursory investigation showed that there was probably another several million in fines & compensation, outstanding in the complaint, that she would have to pay.

We "settled" with an official apology, an official write up in her permanent record, & a change in policy. (She quietly "left" shortly thereafter.)

If you have to track your hours to be paid (or not terminated), then you are by definition not a salaried employee: you are hourly. The way the FLSA works is it asks how the employer treats the employee — if you're treated like you're salaried, then you're salaried. Legally, there's no notion of 'salary' and 'hourly' outside of this. (Ignoring all the real complexities of the law.)

Requiring an hourly worker to falsify statements of hourly work is a serious crime. The repercussions are pretty crazy.