Reading back from the beginning of this thread... you've never stated which model of solar charge controller you have, nor which model of inverter.
Does your new battery have Bluetooth? I presume that's how you are knowing the SoC percentages. Will the BT tell you the charge current flowing into the battery? I would agree if you're keeping the converter online then it should be powering the fridge while the SCC is topping off the battery. How much current is the battery receiving in full sun, if you're able to see that?
The converter you have may also bring the Li battery to a full charge, if you repeatedly cycle the converter power. Resetting the converter will restart a bulk charge timer, and the WFCO bulk charge voltage for lead-acid is high enough to do the job for Li. This may help determine if you can get the battery to 100% for your testing - I'm not suggesting to live with always resetting the converter for normal shore power operation.
The solar charge controller is the Go Power GP-PWM-30-UL. Inverter is WFCO WF-5110RS True Sine Wave Inverter. The battery has a readout display which is how I am seeing the SOC and Voltage. I can’t see directly on the battery if anything is flowing in but I do see on the SCC that its pulling about 9amps in from the panel in full sun so that appears to be working.