Simply put, you are not required to state your name and cannot be denied entry to the secure area if you do and you are not interfering with the screening process-no matter what the TSA says.
Simply show them a copy of this:
CFR 1540.107
http://www.law.cornell.edu/cfr/text/49/1540.107
(a) No individual may enter a sterile area or board an aircraft without submitting to the screening and inspection of his or her person and accessible property in accordance with the procedures being applied to control access to that area or aircraft under this
We do this by presenting our I/D and BP
and specifically noted in subsection b1 and c
(b) An individual must provide his or her full name, as defined in § 1560.3 of this chapter, date of birth, and gender when—
We do this by presenting our I/D and BP
and CFR 1560.3 defines a "Verifying Identity Document as
Verifying Identity Document means one of the following documents:
(1) An unexpired passport issued by a foreign government.
(2) An unexpired document issued by a U.S. Federal, State, or tribal government that includes the following information for the individual:
(i) Full name.
(ii) Date of birth.
(iii) Photograph.
So with all of that, there is nothing that says one is required to state their name