Your inverter photo is showing an input voltage of 12.2V. For a LiFePO4 battery that is 10% state of charge. P.S. There can be a voltage drop under heavy loads but a lithium battery should be able to handle 50 Amps easily. Double check that the charger is set to Lithium battery type.
Your solar setup might be enough if you can heat with the dump load once the battery is full. Use as much passive as possible : good insulation + southern exposure + heat sink + battery heater when you use the space. In my basement I swapped a 500W baseboard heater with 400W underfloor heating...