That's a plausible reason and probably the right one. If you found the episode and paused the screen, you can see the 1880 record Coffey presented had the mother as Catherine and the father as Agustus. Also, the census taker, like you said, probably wrote down the name as he heard it - Catherine Dolan was also Irish, which meant she probably had an Irish accent - so Fred E., Freddy or Freddie would sound similar in an Irish brogue.
I wonder if Coffey actually does know - if you found that record on your own, then one could doubt that he has. I doubt even A&E knows. Someone, especially the mom needs to know they, and her little girl, are stepping in some major woo woo.
The mom and dad, atleast, need to know this other record exists, or her little girl and Chip Coffey are just going to keep at it.