We have had several enquiries about this but unfortunately have not been able to satisfactorily answer the question. As it was a privately owned school it seems that the records were never deposited in an Archive.
This photograph of the school at the very top of Arundel Road on the corner with Cecil Road was taken pre-war. At about this time the school was owned and run by Donald Hughes MILLER who later married Evelyn JACKSON- BARSTOW - taking the name of MILLER-BARSTOW. In a Street Directory D MILLER is listed at Kingsholme, 17 Arundell Road, but in 1940 he had sold it to James Harold MOSEY and bought Etonhurst another well known private school in Weston.
There is a photograph of the Kingsholme school staff and pupils included on the website of King's School Sherborne because apparently when Kingsholme was sold to J H MOSEY it moved to Sherborne House, Gloucestershire and became King's School.
Some names for the period you are interested in are mentioned on the web site for which I've given the link above.