Bit of a breakthrough!
I requested a clearer copy of 1871 Census for Edward McArthur, and noticed that the three step children's surname, which I though was Brown, and has been transcribed as Brown, could actually be Provan.
Since then I found Elizabeth Provan in Airdrie, and her mother Elisabeth McArthur (Maiden name Clarkson) was married to a John Provan before her marriage to Edward McArthur.
My thanks to David Nicoll for the suggestion.