Yeah, I think that Kuvempu Nagar is a good locaclity. But if you are looking for a quiet locality with a lot of greenery, you can consider Vijaya Nagar or the Bogadi area. The Vijaya Nagar and the Bogadi area are a little outside the main part of Mysore city, but both these localities are very calm and peaceful. They are also very close to the industrial hub of Mysore with industries like Kirloskar, and JK Tyres having factories near the area. The Bogadi Road is an important road in the Bogadi locality, and the Vijaya Nagar has the ring road and Hunsur road running near the area. In Vijaya Nagar you can find apartments for rent for around 8,000-9,000 rupees per month.
There are actually a few localities where you can rent out a property for under 10000. I think the best locality for you could be Kuvempu Nagar. The locality is one of the most popular areas of Mysore. You will find many shops, schools and hospitals in the area, and the locality is only 5kms from the heart of Mysore. The locality has wide roads and all important amenities are close to the area. You can rent a 3BHK apartment for 7500 to 10000 monthly rent. You can reach other parts of the city quickly using the outer ring road and the Mananthavadi road which are adjoining the locality. You will also have to pay a deposit before renting a property and you may also have to pay a maintenance fee.