I have an ancestor who was married under one name,
1822 and went back to the church the following year with first child and was using a different surname. He was baptised under the first surname, as well. Some time later they had a child born within the registration period, so we got the mothers maiden name and worked back from that, and found the marriage with the different surname. It was further research , which established the ownership of a property which revealed that his parents were not married, and a statement by a former employer which gave the "alleged fathers surname" as the one which was used at the baptism.
I have since worked out the mother, and traced her line, but with no Christian name of the father am completely stuck!!!!

In your case it is much earlier, so would suggest you look into the other name which has turned up as well. Sometimes you have to go sideways. See if this couple had other children etc.