This is what Reverend Caesar Caine, Church of England Vicar at the nearby St Leonard's, Cleator (1910 - 1922) and author of the definitive history of Cleator & Cleator Moor, wrote about the Catholic chapel and the two schools in this part of the village.
On St Bega's Chapel and the Catholic School, first built in 1853:
"The Reverend Gregory Holden, O.S.B., founded the Roman Catholic Mission at the foot of of Todholes in 1853, building a small Church with School in the rear. When the present handsome Church was erected in 1872, the original Mission Church was utilized as the main Schooloom."
On the Council School (now Grove Court) built in 1910:
" The latest development in connection with Cleator Schools is the erection of the Council School, which replaced the School at the Mill in 1910. ..... The School is a handsome structure standing on a site of 2 1/2 acres on the main road near The Flosh. The site is well situated opposite Dent, and is divided from the high road by a plantation."
Caine, C. (1916), "Cleator & Cleator Moor: Past and Present".
As previously indicated, what is now the Grove Court Hotel was not part of the Catholic church / school grounds and built in 1910 (which is why it does not appear on a map of 1899 but does appear on the 1925 map, Geoff!).
There is also a description of St Bega's in the 1861 Cumberland Directory & Gazetteer which, if nothing else Brad, may at least give you a mind's eye picture of how it looked:
"The Roman Catholic church, built in 1853, is a spacious, but plain edifice, with a low tower of red freestone, and affords accommodation for 500 hearers."
A link to the present parish website was given in an earlier posting to this thread, Brad. You will find contact details on there if you wish to make a direct request to see if anyone has a photograph or sketch of the former building when it was a school.
Attached is another recent photograph of the church grounds (again with the Grove Court Hotel the other side of the wall). This is what would have been the original Catholic cemetery for the district. Unfortunately I do not have the date of the Calvary sculpture in the middle. Some of the headstones seen in the picture date from the latter part of the 19th C.