They could have asked others to issue a license (although in violation of Kentucky law)
Ky. Rev. Stat. 402.050(1)(b) Marriage shall be solemnized only by ... Justices and judges of the Court of Justice, retired justices and judges of the Court of Justice except those removed for cause or convicted of a felony, county judges/executive, and such justices of the peace and fiscal court commissioners as the Governor or the county judge/executive authorizes
A journalist should ask the homosexuals why they insist upon a license from this person and not from any of the other authorized persons.