If you have only 4.5A charging or 12x4.5=66W while operating say a 1500W load , the batteries are taking most of the load and it won’t last long as you can see by the Voltage below 11.5V which is typical <10% SoC ( state of charge) for a car battery and it will start getting sulphated and increase ESR on the plates.
The marine battery can go lower but you need bigger or better batteries and bigger Solar Panels for this load.
The battery voltage drop with the inverter on and load off will tell me something about it’s current. Or the dV/dt voltage drop per minute, which = Ic/C for current and battery capacity in amp-minutes converted to Amp-hours.
An Ammeter would be wise to use or two of them, one for charger, one for output load suitably rated for max current.(either digital or analog)
By computing Battery \$ESR = \Delta V/\Delta I \$ you may correlate this with specific gravity s.g. which are good things to measure health of the battery. ESR must be high, if s.g. Is low in just one cell. This indicates near end of life unless managed better with desulphenator pulsers and proper maintenance and Solar design. Then load R equivalent = V/I into battery ESR ratio determines the voltage drop ratio from no load. I.e. if voltage drops 10% then your battery net ESR must be 10% of the equivalent load resistance (inverter with load ) on the 12V bus.
Cables and terminal resistance will add to this battery ESR (Equivalent Series Resistance)