Would be easier to answer if the duration was known. However, you can cover these places and tweak your stay duration based on overall time you have in hand.
1. Mysore (cover zoo, palace, St. Philomena's Church and Garden)
2. Nanjangud (temple)
and now comes the best part. Drive thru the peaceful Bandipur forest. We saw many deers and Elephants on our way. Never ever stop to click pics . It's not recommended but you can drive at a slower speed to enjoy the beauty. Check the timings of ths road and night driving is not allowed thru Bandipur to avoid animal accidents