We did a day trip to Bandipur recently. We tried getting bookings at Jungle lodges and the other big resorts but could not(luckily for us). Most of the resorts are outside the sanctuary(on dusty roads) and did not look like great places to stay(including Jungle lodges). The prices are atrocious and plainly not worth it.
Best option is to stay at Mysore and drive down. Suggested schedule
Start at 01.30 pm
Reach Bandipur at 3.30 pm and do the safari(35 minutes, dont expect too much)
Do the elephant ride(about 4.30-5pm, 10 minutes)
Start back at 5.30 pm and return to Mysore at 7.30 pm
A similar scedule will work if you are driving from Ooty side
The roads on the last 8-10 kms before the sanctuary starts is pretty dusty and quite bad. Weather is defintely cooler inside the sanctuary.