I think you are a bit confused about what does what? An all-in-one unit incorporates a charge controller, an inverter, and possibly also a charger, all in one box. So, the charge controller section of the box takes the raw solar panel current, and transforms it down to the current voltage the battery wants to see.
Then the battery feeds the inverter section of the box, and converts the low DC voltage into higher AC voltage. Some AiOs make just 120VAC, while others make split-phase 120/240VAC.
The reference to the "idle consumption" refers to how much power the AiO consumes just being turned on but not doing anything. Some are quite high, in the 100W per hour range. Over a 24hr period, that adds up to 2.4kWh. So, your solar panels need to be scaled both to supply the power your loads need, and also the power the inverter itself needs. In comparision, I have 6800W Schneider inverter which only consumes ~20W per hour.
Inverters can be divided into two big classes, high frequency (transformerless), and low frequency (has a transformer). The LF units tend to be large, heavy, and have VERY strong surge capability. My 6800 can surge to 200% for 60 seconds. HF inverters however have little or no meaningful surge. They might advertize 200% surge for 20miliseconds.
The detail you want to pay attention to is the surge of the refrigerator you want to power. I'd guestimate the starting surge to be at least 3-4-5X the running wattage, so a frig that consumes 400W while running might consume 2000W at startup. You don't have to guess at this. You can buy a clamp meter that has "in-rush current" capability. I can measure the startup surge that happens in the first 0.5 second or so. I have this one.
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You can still run it with an AiO, you just have to make sure the inverter can support the in-rush as if it was a steady load, rather than the surge.
I would not spend extra money on an expensive DC frig. They are very over-priced, and if there's a problem, there's no local shop that can fix it. You are better off spending the extra money on a better LF inverter that can run a standard HomeDepot/Lowes refrigerator. At my own cabin, I got rid of all the DC powered stuff years ago, and I'm very happy now living a completely 120VAC lifestyle.