Right on, you need a new 3-4 stage converter of at least 45 - 80 amps.
If charging w/ a 1000 watt generator, you are limited to a 45 amp converter/ charger. If going bigger than that, be prepared to rewire all the way to the batteries.
And yes, turn off the breaker that powers the converter when running on inverter.
And yes, an AGM battery is a sealed, valve regulated lead acid battery, that is capable of venting under extreme pressures, normally non existent under normal use.
And no, the converter has nothing to do w/ charging from the alternator. It charges directly from the nator, and I see no problem using the nator as your major source of 12v charging