There is no fundamental reason why it is more difficult to charge a 12v battery than a 6v one. Voltage doesn't come into the equation.
You seem to be struggling by choosing some motor at random that doesn't suit your 6v condition well and suits your 12v condition even less. Unless you are willing to use a more suitable generator then you are forced to use inefficient methods of adapting your 6v to higher voltages.
You can use voltage converters but at 6v you need decent circuits to avoid big losses from diode drops. It may be your only way unless you are prepared to break away from the unsuitable high speed motor.
Your 20:1 gearing is crazy and it was as a joke that I suggested increasing it to 40:1. Even then it may not be much worse than boosting the 6v to charge 12v.
With a suitable efficient low speed alternator and perhaps a modest amount of gearing you could improve the results immensely. In fact I doubt that the gearing will ever let you get the best results but it may let you get there more cheaply rather than using more expensive magnets.
If you must work from 6v then you have the options of a boost converter ( which for best results would need to be specially designed) or you can use two 6 volt batteries in series and devise a way of connecting your charging power alternately to each and if you incorporate some voltage monitoring scheme you may be able to keep them reasonably in step so that each maintains a similar state of charge. You could charge the batteries in parallel and discharge in series but in that case you can't use power as you generate it and you would need 4 batteries and work them in pairs alternately, charging 2 and discharging the other 2.
Flux
