Hi Andy,
Searching on Ancestry for school* as part of the address in Civil Parish Inveresk in 1891, there are two schools that come up:
Loretto School
Windsor Park School
Loretto School seems to be the biggest and is residential. Lots of students given on the census.
Windsor Park School has only the headmaster and his family, an assistant teacher and some servants mentioned, so it was probably a day school.
So far not found any others, but "school" as part of the address may only give the sort of schools you don't want... But at least it got you one more name!
All the best,
Sarah
Later: Still trying, this time searching for anyone with occupation "teach*," there are 67 results, most of which don't give any indication of what school they are teaching in.
Jessie T Brown is an elementary teacher living at The School House Old Craighall, but the only other people with that address are a Pit Banksman Labr and his family...
In
1901 there is another school: New Street Newfield Boarding School
Jesse D Dunn is a teacher with address "Schoolhouse Cowpits" (great name!

)