As it is now the law of the state in the USA and elsewhere in the world to issue marriage licences to all concerned, it is the state that should accommodate all by having one clerk serve only traditional heterosexual couples, and another clerk serve only non-heterosexual couples. (I say non-heterosexual so that identities such as trans-gender, bi-sexual, pan-sexual and poly-sexual can be included in the definition.)
Before anyone objects with the argument that this solution would hail back to segregation, being white or of any colour or ethnic background is not the issue here. Segregation was separate buses/divided bus sections and other visible dividing lines. The hypothetical waiting lines, for same-sex and traditional couples seeking marriage licences in the one and same government office room, would be a choice of preference. (Such as preferring one gender of public powder room over another.) Any violation of crossing an (invisible) line to cause a disturbance would be breaking a state law, and not infringing on the clerk’s beliefs standing behind the service wicket, their beliefs already protected by the same law.
Lorraine Dmitrovic: The Surreality of it All.... Film, books, the arts 