Apologies, if I didn't make myself clear: I meant that it is impossible to conduct a same sex marriage without endorsing homosexual activity. In contrast, it is possible to employ or provide medical treatment for them without doing so.
The relevance of the registrar's endorsement can be illustrated by two people who are closely related or underage wanting to be married. The registrar would be sanctioning sexual intercourse. Maybe sanctioning/giving permission is a better word than endorsing.
Then you would be in the wrong job.
To be a registrar, according to the careers service:
You’ll need:
the ability to relate to people from all backgrounds and cultures
tact, patience and empathy, for dealing with people who may be distressed
the ability to understand and apply rules and laws
clear and accurate handwriting
the ability to work under pressure
administrative skills
So, to sanction illegal marriages, is not part of your remit.