Richie: SJLibrary.org (the San Jose Library catalog) has an entry for "There's No One Like Mother To Me!" (no date listed), with "Davies, Charles" given as both composer and lyricist. And you can find a couple pieces of music by Charles A. Davies at the Levy site, dated 1885 and 1886, though not the piece in question.

But Gussie Lord Davis is clearly credited with writing both words and music for the 1885 "There's No One Like Mother To Me", and he was famous for writing tear-jerkers like this. The similarity of his words to the Carters' version is undeniable; I haven't checked the tunes. It's improbable he first wrote and published the song as "Davies" in 1877, when he would only have been 13 or 14. And note that Davis's song title lacks an exclamation mark, while Davies' has one.

So it appears there might be two songs with the same title, written by composers with similar surnames. It would be interesting for someone in San Jose to compare the library copy against the scan of Davis's song.