I have 6kw of inverter, 5500w of PV and something like 20kwh of battery. That setup runs a critical loads panel. Currently its in the mid 90's and the AC runs for about 18hrs. (the furnace/blower is run off the solar for winter heat, but not the outside AC condenser)
So a hot day looks like the first image, with an avg use that hovers north of 1300w continuous (due to the blower motor) on a cool day in the spring it looks like the second. not only do the panels peak higher watts because they are cooler, I also only avg about 600w usage (or half as much)
in the spring I can go several days off grid if sun is good. Currently I can make it till about 3 or 4am then transition to the grid (out of batt) I know I'm not hitting 100% SOC on my batt because i dont have enough PV to charge it.
TLDR, at 5500w PV I'm short on my PV to charge my battery fully in the summer and the amount I make won't sustain nearly as much load as you have. So yeh your short a LOT of PV IMHO.
Note: attached data is only 1 of the 2 inverters (they are parallel and share the load) so take everything you see and 2x it.