The charging/12V power supply (your converter) pictured is a ferroresonant charger. They usually have only one set point voltage for charging and supplying power. It is a poor device to use with AGM batteries since the lower set point will mean that your new AGM's never become fully charged when in use.
If you're going off-grid for a certainty, I would suggest to definitely look into solar and that may allow you to keep the older converter for shore connection and occasional battery charging. Solar, with a good controller, does a great job of charging and maintaining batteries. Most good controllers have adjustable set points, some with temp compensation so the batteries are treated as well as possible. AGM's aren't cheap and anything you can do to keep their health is an efficient financial process. If you want to wait on solar, I would suggest to add a modern three stage converter. You can disable the old converter or interrupt the power with a switch or parallel it with a new converter. The latter might be a good solution since most AGM batteries love to be fed lot's of current.
