I don't understand the bit about "her office's policy of not issuing marriage licences to any couples, gay or straight". What's her problem with heterosexual couples marrying? Was she elected to her post before or after same-sex marriage was legalised?
		
		
	 
She was in office before the decision.  County clerks are elected to multi-year terms, just like mayors, senators, congress people, and the President.
Basically, she chose to throw the baby out with the bathwater - "If marriage won't be confined to just one man and one woman, to the exclusion of everyone else, then you know what?  We'll just make it so that NOBODY can get married here - straights OR gays.  We'll just stop issuing marriage licenses altogether, and that'll fix that!" - that's essentially where she's coming from.
Although, that logic is absolute ludicrous, given that the issuing of a marriage license, thus allowing two people to get married - that is a courtesy service of the government.  Just like the postal service is a courtesy service of the federal government.  The issuing of a marriage license is a courtesy service of the government.
I use the term "courtesy services" as opposed to mandatory or emergency services that the government HAS to provide - police, fire, 911 service, a medical facility in every county that's been designated as the county hospital - those are all mandatory services of the government.
The fact that you can go before a court judge, a justice of the peace, or the city mayor in any town in this country and actually get married - actually have them officiate the marriage ceremony - those are courtesy services of the government.  While not absolutely necessary to protect life and property like police, fire, and 911 services are, marriage licenses and the ability to go before these people and have them pronounce you as married - those courtesy services are still services of the government.
And the government has the obligation to make those courtesy services available to anyone who meets the legal requirements to obtain or use those services.  If, say, Mike and Greg both just turned 18, and just graduated from high school, and now, as legal adult men, they've now decided that they're in it for the long haul, and want to get married, the government is required to provide the courtesy service of a marriage license.  And if Mike and Greg want to pay the fee to go before the county judge and say, "I do", 
the government has the obligation to provide that service to them - irrespective of the personal beliefs of the person performing that service on behalf of the government.  It doesn't matter what the personal beliefs of the county judge are, if Mike and Greg go to him or her, and pays the fee for the service, that county judge HAS to do it.
It's the same thing here with Ms Davis.  The government doesn't give two good fucks what her personal beliefs are - these people are coming to your county and asking you for a marriage license.  If they have the money to pay for the license, the government's position is - shut up, smile, take their money, and give them what they've paid for.
I will say one thing here, being in an open-to-the-public job myself - it doesn't matter who your customer is.  You have a responsibility to make it easy for them to obtain the good or service they're looking for.  You can't deny the customer a good or service simply because that customer is gay, or speaks Spanish and only very little English.  You can't deny the customer simply because they're a woman, or black, or has blonde hair, or whatever.  You simply cannot do that.  You have a responsibility to, as best as you can, meet the customer where they are, and cater to their needs.  If it's a gay couple applying for a marriage license, or wanting to actually get married, catering to their needs means, in this case, that you don't care that it's two men getting married - you issue the marriage license, anyway.