I havent tried this but I doubt the stock head unit and the steering wheel controls would work in harmony with your bluetooth adapter. Perhaps with an aftermarket headunit that had bluetooth built in and wiring that up to the steering wheel controls could work.
Try calling a company like crutchfield and see if they can recommend something, i know they have few bluetooth ones for less than $100. They also should have an adaptor to get the steering wheel controls working too. Not to mention, pretty much any aftermarket head unit will sound better than stock, especially if you were using an fm modulator to play audio from your phone to your car.