Having tried to make sense of this am not sure re connections.
In Cheshire BMD, Catherine is the only one with mmn Moore - other children are Maher.
1911 information re Michael and Margaret Lonergan does not fit with numbers of children to include Catherine.
Looking at past censuses, Catherine Chappel seems to be Catherine Moore/s and we have the Chappel connection already mentioned so in part, I think there is a connection but not yet sure where.

There is a marriage
Chorlton September 1889 vol 8c pg 1169 Michael Lonergan and Mary Ann Moore