OK, so you're not really measuring 'Solar Production', you're measuring just what your house is using from either:-
a) direct solar production if house load is load is less than available PV power
b) house usage from battery, but have restricted a smallish battery to only 50% DoD
Then you will need to consider :-
1) you will be drawing from the grid when your house load becomes greater than what the inverter and conjure up from PV and battery
2) you will be drawing from the grid when the battery is depleted (and at 50% DoD on a 10kWh battery we're probably only looking at between 4 and 5 kWh of available stored energy)
and
3) if the battery is full (or nearly full) and the house is not using much, the inverter will ramp down PV production to meet battery + house demand.
Now that it is October and you're probably not producing anything between about 4pm one day to 9am the next day _and_ it's probably only 60% of 4.8kW when the sun is shining this time of year, then I don't see there is necessarily a problem
It sounds to me that you need to have more finer granularity of measurement and log not only AC output, but PV generation, battery charge, discharge, SOC and grid usage - I do that with my grid-tied system using an emonPi with 5s resolution. You can then clearly see what is using power, when and from whe