Not big on Auto electrical, but guessing your son has gone back ad disconnected it all, and checked the batteries voltages, then added a bit at a time and checked voltages again, to help verify the readings you see on the controller.

Check the instructions on the inverter, it may need to be earthed.

Cables don't seem that big, 12 volt systems tend to run larger cables, (look at the cars own earths.)

Suggest a visit to an auto sparky.

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