The 12 volt supply ain't an issue. Replace the extra lead battery with a LiPo pack of the same weigth, and a 6s psck is cheap and rechargable with a equaly cheap and light charger that happely will use the on board 12volt supply.

But yes, you're right Nick, a normal linak is artritic slow for this purpose.

With only 3 positions I see a solution with 4 solenoids.
How fast is an Arduino? The speed of a mosfet is known and more than fast enough.
This also solves the sync issue.