Looks like the main problem was that the circuit after the battery had some periodic current spikes, probably because of the switching voltage regulator. The average current was 20 mA, but the spikes were about 200 mA (1 mV = 1 mA) :
I added a 100 uF capacitor in parallel (ceramic X5R, 1206, 6.3V) to the battery, and now it looks like this:
and it charges up to 4.18 V instead of just 4.02 V. New charging curve:
Is it safe for the charging IC to use such a big capacitor? Can I even increase it?
It still doesn't stop charging, but this is understandable, because the configured termination current is 6 mA, but the circuit needs already 20 mA. I will try to increase the termination current, and reduce the current consumption of the circuit when charging, then this should be solved.
