This construction has the grammar elements turned around and out of order. The word "that" is the direct object, which has been pulled out to the front of the sentence. If the direct object were longer, it would absolutely have to have the comma.
...The name of the man who waited on me that night, I don't remember.
...The name of the man who waited on me, I don't remember.
...The name of the man, I don't remember.
When it gets down to just one word, the comma is based on whether you think it could be misread.
...The name I don't remember.
...The name, I don't remember.
I think a comma here works best so that the meaning is clear.
In the example being discussed, because it could be a "that" clause without the comma, I think it should have the comma -- again, to prevent misreading.
Happy punctuating!
Margie