How to run a fridge on off grid system
The inverter itself if it is thought for offgrid application should have the undervoltage protection. So you can connect the battery directly to the inverter. This is how it is normally done in off-grid solar appplications and the MPPT charge regulator is only for the DC loads and regulating overvoltage while charging. Otherwise you would need to always oversize the charge regulator.
See an offgrid circuit schematic of one of my projects