Does it have to be a hotel? We're headed back there later this year and, like last time, we will rent a private apartment, this time through AirBnB (we used Homeaway last time).
A car is useless in Bellagio. It is quite compact, and besides, tourists cannot take their cars into the old quarter, which is where I would stay. The only issue you will have self catering in Bellagio is obtaining groceries without a car. There is a very small convenience store in the old quarter, but to reach either of the two larger stores, you need transport or to use the bus. If you can't get around this, a hotel is your only choice but all the good hotels in town are fairly expensive.
As much as I love Bellagio and prefer it to Varenna though, I wonder why you're going to stay there. It is a very short ferry ride from Varenna so, if you've already decided to stay there, why not just extend your Varenna accommodation and make day trips to Bellagio?