We are getting there slowly.
I still think there is more problem with your measurements than with the alternator.
How do you measure 22v when connected to a battery. Either you are measuring the wrong thing or the battery is sulphated and not holding the volts down.
The battery should hold the volts down to 14v or less for hours at 2.2A. You don't still have that lamp in series do you? if so take it out and try directly into the battery.
I don't know if you have 16 or 18 magnets, you mention both. If 16 you would be better off with 12 coils and make it 3 phase. I am sure you can get your desired 10 A without laminates and a fair bit more with them. You already seem to have a low cut in speed for the output you are expecting so with more coils of fewer turns of thicker wire you ought to get the amps up quite a bit.
Before you change anything sort out your measurements and find out what it is really capable of, then you will have useful data to make changes. If you add laminates at this stage your cut in will be very low and if you are measuring the wrong things you will confuse yourself.
So far 22v 2.2A at 150 rpm is not bad going for a single rotor air gap alternator with only 8 coils.
I assume you are using a single phase full wave bridge rectifier not just a half wave diode. If so try again directly into a good battery, make sure the dc volts hold down below 14 and the ac volts on the input of the rectifier are similar or lower to prove the bridge is not half open circuit. See what amps you get into the battery with different speeds.
Good luck
Flux