If you provide a service, you should not be able to discriminate.
Or, on other words,
If your religion prevents you from doing your job, find a different job. If your religion prevents you from being a kind and decent person, find a different religion.
And I would extend that to being “a merciful and empathetic person”, as well.
I think you need to review what that oath entails. Because it does not include not killing someone. It is about doing the least amount of harm… and there are some harms that are much worse than death.