You could try contacting Pates to see if there might be a car share option? I know they have advertised on school website etc. to help arrange these in the past. Sometimes that works well as if halves the driving and allows DC to go to school of their choice
Most public buses do not go up Princess Elizabeth Way/Hester's Way (past Gloscol) and DC would have to walk from bus drop off to Pates. Quite a long walk with heavy bags. I suggest maybe you could try public transport journey to get the feel of it or ask around your primary for another DC who might be going to Pates. The private bus is expensive and may not go near your area.