Inverter Wiring:
2000W/12V=166 Amps. I would use 175A fuse between the inverter and the battery with
2/0 AWG wire. That should keep the voltage drop at 3% as long as the wire is less than 10' long.
PV Wiring:
Assuming the PV panels are in parallel, 500Watts/13.5V=37A. Multiply by 1.56 per NEC to accommodate certain reflective conditions and you end up with 58Amps.
A #10 PV wire is 55 amps, which is close enough, assuming you stay under a 15' run. I would use a 60A fuse at the PV controller.
So you were pretty close...
Bruce
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2/0 cables