I would like to thank you for your help. Finally, I have solved the problem. I concentrated on waking up the device as you mentioned.
When I pushed the action button at least 2 seconds there was no reaction (this is the reason why the previous log did not contain WAKEUP_NOTIFICATION message). So I decided to try to heal the device (from HABmin) after that I again used the action button in wakingup mode and I noticed that the device started communicate with the controller. I had to wake up the device several times but finally
he started initiating transmissions to the controller himself. Now it works without problem.
I think there were the problems with unfinished initialization although in the PaperUI I had online status.
Thank you, once again