For public use you need not have a TC at all but you must be a public agency, on the governmental mission for which the aircraft is required. Experimental was not how this aircraft was listed in the registry, it was restricted. Of course that could be out of date, it isn't like the FAA is on top of that all the time but
even if it were experimental the following applies as regards uses of the aircraft that either has no type certificate or does not conform to it's type certificate.
It doesn't sound like this flight was any of these.
Research and development: to conduct aircraft operations as a matter of research or to determine if an idea warrants further development. Typical uses for this certificate include new equipment installations, operating techniques, or new uses for aircraft.
Showing compliance with regulations: to show compliance to the airworthiness regulations when an applicant has revised the type certificate design data or has applied for a supplemental type certificate or field approval.
Crew training: for training the applicant’s flight crews in experimental aircraft for subsequent operation of aircraft being flight tested in type certificate programs or for production flight testing.
Exhibition: to exhibit an aircraft’s flight capabilities, performance, or unusual characteristics for air shows, motion pictures, television, and similar productions, and for the maintenance of exhibition flight proficiency.
Air racing: to operate an aircraft in air races, practice for air races, and to fly to and from racing events.
Market surveys: to conduct market surveys, sales demonstrations, and customer crew training for U.S. manufacturers of aircraft or engines.