The yield from the north is completely irrelevant. Max it out. The [Sowiesokosten] make purely south-facing occupancy far too expensive compared to if you let everything be covered.
Current example (all in offers) for a system in Stuttgart from 10.04: 7.2kWp south-facing occupancy + 10.2 kWh LFP battery: €15,999 (= €2,222/kWp incl. storage) 13.95kWp south-/north-facing occupancy + aforementioned battery: €19,499 (= €1,398/kWp incl. storage)
I wouldn’t even begin to think about the yield in the north. With the extra cost of €3,500 for 6.75kWp (= €518/kWp), the upgrade always pays off.