I have no religion, I'm beating a civ about to get a religious victory in combat - how am I able to hamper their religion?

The first thing you can do to slow down the religious advance is to start targeting the AI religious units. While at war, military units can attack and destroy enemy religious units without taking damage. This is a good way to prevent the AI from spreading their religion to even more cities.

If you have any city that is following a different religion, start buying Missionaries and Apostles to spread that religion in an attempt to counter the spread of the majority religion.
The better option is to buy Inquisitors, if you can. Use them to remove the majority from the cities that are following it. That will be one less city following the near-victory religion.