....As far as when it would become illegal, the only time that would be enforceable (and hold up in court) is if they actually sighted the airplane flying, and waited for you to land, to taxi it up and as you stepped out, and tapped you on the shoulder.
And what, at any other time, if an agent merely asks when you installed the prop, if you flew with it, checks against the airframe lots, etc.? Lying is a felony, even when there is no underlying crime (ask Martha Stewart):
18 USC 1001Except as otherwise provided in this section, whoever, in any matter within the jurisdiction of the executive, legislative, or judicial branch of the Government of the United States, knowingly and willfully
(1)falsifies, conceals, or covers up by any trick, scheme, or device a material fact;
(2)makes any materially false, fictitious, or fraudulent statement or representation; or
(3)makes or uses any false writing or document knowing the same to contain any materially false, fictitious, or fraudulent statement or entry;
shall be fined under this title, imprisoned not more than 5 years or, if the offense involves international or domestic terrorism (as defined in section 2331), imprisoned not more than 8 years, or both. If the matter relates to an offense under chapter 109A, 109B, 110, or 117, or section 1591, then the term of imprisonment imposed under this section shall be not more than 8 years.
Taking the 5th in what should be an administrative procedure will be awkward.