The size of a PM-generator should be chosen such that there is good matching in between the windmill rotor and the generator.
While we probably have different opinions and advice as to how you might proceed, I agree entirely with Adriaan here.
Broadly, if you are direct-tied to batteries, and have too much prop (rotor area) you start out good in low winds, but run the risk of burning out the alternator in high winds. On the other end of the spectrum if you don't have enough rotor area, you never really get the blades 'flying' where they are most efficient. The rotor wants to follow a cube power curve as suggested, but the alternator without electronics (MPPT, or at least change in line resistance) pretty much wants to rise in linear manner relative to speed. The result is we are forced to try to find a happy middle where the load match is reasonable - but there are compromises at best.
To complicate thing further, we spend an inordinate amount of time on theory, blade profile and coil configuration and the like, only to find out the load is also changing as the batteries are charged, loads are applied, and or other charge sources (PV) are introduced.
how can i get a lower cut in speed?
Stronger magnets is one way. You've got that... but, if you look at some of the designs that Hugh, the Dan's and others have worked out, I think you'll see rotors in the 7'-10' range with comparable magnet/coils to your initial proposal. I would look to those as a guide, and either increase the rotor diameter, or build a less robust alternator. They've both got some excellent books on the subject. There are also several blade calculators - Hugh's, Alton Moore's (and, I've only glanced at Adriaan's compilations).
Early on you stated the project was "for fun". I'd say jump in, and have at it, but use some of what's readily available.
~ks