I'm interested in doing something like this as well. Build a pergola with a solar roof instead of just a metal one from Costco. I've seen a few videos online doing a similar sized system with like a Jackert or Anker Solix. Realistically with 2-3 harbor freight panels they're only enough to power like your home office. A fridge will burn through the battery pretty fast. I do believe you can have them do input and output straight into your wall outlet and you don't have to plug right the appliances into the battery/inverter.
I agree, a solar pergula would be good and hardly noticeable.
A fidge is only going to use 1.5-2kWh per day. A medium sized pergola would give you more power than that. Since you aren't opening and closing the fridge in the middle of the night, a 1 kWh battery would keep it running all day on normal days.