I have some new information. I reat that only an external charger should load the extended battery over 3000mah.
Bought a 7500mah battery and charged it over night with BATTERY INSIDE THE PHONE. The battery indicator goes a little weird and jumps sometimes, maybe this calms down with some charging cycles. Also I made a battery calibaration after the first full charge at 100% even though some say that this is not necessary. This time I had "only" 6 hours SOT.
Bought this one. After a week of full charge and discharge cycles when the phone is off, the USB tester showed 5000 mAh. Slightly less than stated (7500). But at least he keeps it without any glitches. It will be necessary to try to charge the battery separately with external charger.