9V battery for RC car

So walking through walmart today on break I happened across a 4x4 RC car (truck really) and I decided I needed it.

It takes 6 AA batteries and testing with the meter determined that they're in series so it takes 9V, I would like to add a 9V rechargeable battery with a larger MAH capacity.

The space I have to work with without cutting plastics (Which I can do) is approximately 3"x3"x1".
Any thoughts?
I could use a 12V rechargeable battery and step down the voltage to 9V, and use a standard 12V charger to recharge.

 

I could, but with some minor modding I could fit this, charge it with a 12V charger, and just have it step down the voltage for the car.
http://www.batteryclerk.com/store/p...m_term=10159&gclid=CLC4xPKhjsgCFYsvgQodBD4JPQ

with this.

http://www.eachbuyer.com/12v-to-9v-...USD&from=pla&gclid=CI-30OCkjsgCFUg7gQodChcOzw

Because I should (In theory) get a higher run time with the the 12V 1.2AH battery stepped to 9V, rather than a 9.6V 2000mAH battery, correct?