The 2.5mm/4mm/6mm cable from the inverter to the CU allows the inverter to send current from the PV panels into the CU, to reduce the current being taken from the "outside connection".
That's all you need, no need for a cable from the CU to the "outside connection".
The breaker in position 3 with the label torn off has a 40A rating, suggesting a high-current device such as a shower was, or may still be, connected to it?
The breaker in position 7 with no label has a 30A rating, the same rating as the "Existing Sockets".
So maybe an extra set of sockets have been added since the house was built, in an added conservatory perhaps?
If you're not sure what it does, switch it off and hope nothing stops working.
Then it may be be a "spare way" you can use for the inverter.