Since the battery and charger are from the same folks, I'd simply ask them which connector is compatible with the battery pack you pick. I'm sure there is one for the charger that will charge their own battery pack. 
Let's say you picked this 2400 MAH pack, it's a decent capacity and fairly cheap. It's charging plug is described... 1.5" wire with 5 pin Female JST plug
I suspect one of the choices for the charger is compatible with that plug. Note that you can't just charge the whole pack, multi-cell lithium packs charge with monitoring of the individual cells. So, it's is important to match the charger connection to the battery charge port.
I'm not sure why Paul has so much trouble with LiPo batteries, my brother is an avid model airplane guy, and he uses LiPo exclusively for all his stuff now. He gets many cycles from those packs. One advantage with the LiPo solution is you can go for a larger battery pack and get hours of running on a single charge, so you'd be able to have an entire session with a single battery and no breaks for charging.
If you are discharging at a 1 amp rate, which is roughly what we are talking about, the AH rating of a primary AA alkaline cell falls to around 1 AH. You'd want at least eight of them to bring the voltage up to around 12 volts for your configuration. Also, fairly early in the discharge cycle, the voltage falls off significantly, further impacting performance. You'd get maybe an hour of running, if that, and you'd be buying new batteries regularly. That will get very expensive fast.
For any doubting Thomas in the crowd, here's the PowerStream Discharge Tests for various Alkaline and NiMh AA Batteries.
You could go with something like a bunch of NiMh AA rechargable batteries in holders, those would be a lot more cost effective than using primary batteries. They also stand up to higher current discharge rates better than Alkaline batteries. Alkaline cells are best suited for low discharge rates.
