I've never tried battery powered paint stirrers. Some of my older paints are way more solid than liquid! Someone suggested one of these. An Ikea coffee frother (300.000.41). Cost me less than £3 on ebay.
It looked like it was a shade too big to fit in MP and old (black top) GW pots but the spring compresses down so it will actually fit OK! Tamiya pots are wider, so no problems.
Did some painting today and had a need to try out the frother. Tried it on Tamiya, Black top citadel and MP paints (and more importantly, the pot size). All worked excellently. Does not throw paint out the pot as I worried. The motor is pretty low torque so it will stop if pressed too hard against the bottom or sides, but in all cases you can find a sweet spot where if effectively blends the paint.