It would be helpful if you name the converter that you presently have.
To answer some of your questions: Yes, you can plug your shore power cord into the generator and use a stand-alone battery charger at the same time. You're only limited by the output of the generator. You don't have to use the charging 12V port on the generator. In fact, most are limited to <10amps. You can plug a stand-alone battery charger into any AC receptacle that is handy.
You would be well advised to connect both your batteries in parallel and leave them like that, both for charging and for use. You gain power by the Peukert effect. There is nothing wrong with using your 12V system while the batteries are charging. It will just take longer to reach an adequate charge.
I would suggest that there are improvements that you could make: using two 6V GC2 batteries will give you more reserve power and are easier to keep in a good state of health. Installing a new, adequate converter will spare the expense of a stand-alone charger and serve you better in the future, resulting in more average time between battery swaps. You should be doing an equalization charge routine with all lead acid batteries. It will result in longer battery life. You don't mention a disconnect switch for your 12V system. You should have one. Increasing the wire size from your present converter to the batteries will result in somewhat faster charging.
Finally, a small solar system is the best battery charging and seamless power plant you can find. If implemented correctly, it will replace your generator, carrying gas, and the noise endured. When the trailer is in storage, no extraordinary methods are needed to keep your batteries charged and ready to go.
Best of luck with your 12V system!
