If this really is the case then you have no wind.

I suspect you are reaching cut in much too early and going into stall. I am not sure what sort of ammeter you have, but try something that will measure low current and check that it is really not reaching cut in. If you get a few mA at very low prop speed then it is cut in.

Alternatively read the ac volts on a pair of the bridge ac connections and it should rise with speed. When it starts to go constant at about 9v or so you have reached cut in. If in doubt remove the battery connection and if you are cut in and stalled the speed will shoot up.

If for some reason you are genuinely not reaching cut in then suspect useless turbulent wind or some alternator fault dragging the thing down, possibly a shorted diode.

If it doesn't get a real rant on with no battery connected then disconnect the rectifier and see if it gets away.

If you can establish the cause of trouble then it is easier to know how to fix it.

I am pretty sure it is stalled and badly so. Rather than increasing gap 1/16" I think removing one magnet rotor would more than likely be needed. That is why I suggested going for 24V.

Gap is a convenient way of tuning the cut in speed when you have it nearly right. It is safer to aim for a slightly low cut in speed so that variables in construction don't bring you above the intended figure. It is not a good idea to aim more than a bit low.

The first step is to get cut in speed right ( prop running a fair bit above intended tsr at 7mph). That puts you in the correct starting point, then if it stalls in higher winds you add line resistance. Resistance will not cure it if the cut in is too low.

Although increasing gap slightly is a safe way to start, if the gap needs to be increased much you are wasting magnet and it would be far better to reduce turns and keep the same gap.  For some reason most people will go in with far too many turns in the belief that it will give them more power and generally it does the opposite. If it is so obviously wrong they question it but I have little doubt that many are accepting a performance well below the possible because they have gone in with too many turns.

You have to get started with tsr above design so that in the low wind range you work through the peak of the prop curve. Tsr falls rapidly with load so if you start below the peak you will be doomed from the start.

In this case from the stall point of view it would stand a larger prop but if it is a good wind area the little alternator may not be able to cope with it unless the furling is very sensitive.

From your description of your site you bring me back to my original suspicion that the real trouble is lack of wind.

Urban sites just don't have any wind, plenty of turbulence to blow things over and rustle trees but with little usable power.

The best chance on a turbulent urban site is with a prop with lower tsr and slow alternator. You may possibly do better with more coarse pitch blades even pvc if you get the angles right. The maximum output will be low but you might get something more often.