Now I measured the standby current consumption without bigger caps, with just one and with two. Cyble module + sensors were powered in stop/shutdown mode. I got 40 nA with no bigger caps and just a 0805 10 uF ceramic cap. Then I added 1206 47 uF ceramic cap. First few seconds the consumption was 200 nA but in a few minutes it was 80 nA and still dropping. Nothing to worry about I think, since even 0.5 uA would be less than I expected, since the sensors should take almost that much, but seem to take less.
Then I notices something strange. The current consumption went to 2 uA when I put my small 40 W work light on with a traditional bulb. By shadowing different components with a finger or a business card I found out it was the BLE module that was sensitive to light. It was much less sensitive to a much brighter LED flashlight (>4000 lx vs. 600 lx according to the sensor on PCB). The sensitive area seemed to be the one with metal shield. How can it be affected through the shield? There are openings to the side.
I got the new batteries. Brand new Varta CR2430 seems to have less than 10 ohm internal resistance. Also it seems IR of lithium coins depends on how it is measured. I have a DE-5000 LCR meter. I measured resistance through a capacitor and the higher frequency I used the lower resistance I got. E.g. for a used CR2032 I get 130 ohm at 100 Hz, 122 at 120 Hz, 43 ohm at 1 kHz, 23 at 10 kHz and 17 ohm at 100 kHz. Using a 470 ohm resistor and multimeter I got about 180 ohm and calculating from rise time after a pulse I got about 300 ohm.
I first thought that DE-5000 + the cap I used fails to measure, but measuring a NiMh AA-cell gave about the same at every frequency (0.4-0.5 ohm).
This battery had been in my Polar heart rate monitor and was dead there. It still shows 2.93 V. It could not power up my device as is, but adding a 330 uF cap made it work OK.