you can do just about anything you like, however not everything that can be done should be done.
your genset, if it can produce 9kwatt at 120vac works out to about 75amps max
that is the limit whether you regulate it to 12volts or 120 or anything in between.
that is the limit of what the stator windings can take without overheating and damage.
so 75amps x 14.4 volts = 1080 watts
so basically the unit will then produce about 1/9th its original capacity should you decide to regulate the output
down to and rectify for charging at 12volt nominal,, not a very good prospect at all.
any engine running at 1/9 its capacity is likely to be nowhere near as efficient as it was intended to be.
so you see, it can be done, but probably shouldn't be done from a theoretical standpoint.
now for a practical standpoint, the regulator you list as a possible controller for the generator is capable of supplying perhaps
10amps to a 12vdc field, it is very unlikely the field in your genset runs at anywhere close to 12vdc, more likely it runs at 50vdc or even
higher, which precludes the use of that regulator for this purpose.
btw, that regulator was designed to control automotive type alternators, such as used on cars and heavy trucks.
you would be miles better off just plugging in a good quality charger into the genset in my opinion.
basically i would not risk damage the an otherwise useful genset, and i know a bit about them to start with.
not a good solution to your problem, sorry
bob g