I don't believe there is any kind of a rule that says you must or cannot use a particular tense with at that moment. It depends mostly on the meaning, and also on the characterisics of the particular verb.
If you are talking about a punctual event, you use the simple past:
At that moment I dropped my keys.
At that moment he appeared in the doorway.
At that moment I realised I was lost.
If you are talking about an action of state that was continuing through the moment, you usually use the continuous:
At that moment, I was sitting at home, waiting for a phone call.
At that moment, he was eating lunch.
[It is not a coincidence that I have used commas in the last two example: "at that moment" is playing a slightly different role in the two cases. With a punctual event, "at that moment" defines when it happens: in narrative terms, this is usually pushing the story forward. With a continuing event, it is generally giving background information rather than pushing the story on].