Yes, all are bound by the Constitution.
The Constitution (as it was written and intended) limits the powers of the U.S. government (and in some instances State governments), not the freedom of its citizens.
If the Constitution is requiring this elected official to service this demented perversion of marriage, then the Constitution isn’t worth spit.
The Constitution applies to the Government.
Not one word of the Constitution would protect a private employee’s job if the employee refuses to comply with the Private Employer’s lawful instructions.
That’s the beginning and end of it. The Governent can’t oblige conduct if that conduct violates a reasonably held religious belief or value. Not so for private employers. If an employer needs employees to serve drinks to customers there isn’t any sane set of principles, certainly no Constitutional ones demanding that the employer hire people incabable of doing the work.