“If we run the generator for 4 hours in the middle of the week we can usually last a week...”
Which takes a toll on your batteries’ lifespan. Fully recharge daily is the best maintenance.
The starting point for necessary solar watts is one per battery amp hour. Four GC batteries would be more than 400 AHs. Two hundred solar watts is very inadequate IMO. The controller offered is a PWM. I’m a fan of MPPT controllers for efficiency. And the 10 gauge wire doesn’t impress me for the system you’ll need. Too much line loss. What will be your wire distances?