Mysore has a few good schools like St. Arnolds Central School, National Public School, St. Josephs Public School and Pramiti School. While most of the schools are far from where you stay, you can enquire about the bus facilities that these schools provide.
Sorry to inform you, but there are no schools in the Bilikere area. The closest options are National Public School and Excel Public School that are all around 8-15 km from your area.