First. I am with everyone else. Mixing battery chemistries is not a good idea. If you look at the discharge curves, lead acid shows a continuous slow decline. Lithium stays flat an drops quickly at the end. Your lithium batteries will constantly be charging your lead acid.
Second, you will likely need 2 devices to achieve your goal. A solar charge controller and an inverter/charger/automatic transfer switch.
Install the new inverter as close to the battery bank as possible. Remove your existing converter and leave the DC distribution panel and wire it directly to the battery bank via a fuse. Rewire your shore power plug to go directly to the inverter/charger. Wire the output of the inverter to the input side of your AC circuit breaker panel.
If the OP intention is to run A/C off of the batteries and inverter, he will need at least a 3000W inverter, more battery storage and a Soft Starter on the A/C.
Buy a small (1000W) pure sine wave invert and plug the refrigerator into that.
Buy a 12VDC compressor refrigerator.
Retired. 31 year of automotive engineering for one of the Detroit 3, specializing in Powertrain Control Systems.