I will contact the developer of the method 2 as you suggested, thanks.
In the meantime something unexpected happened. Today I saw the button registered and working under Xiaomi BLE integration…!
I have only one possible explanation in mind since I am definitely sure I haven’t extracted the key with any method. I have already a LYWSD03MMC thermometer connected with the Xiaomi BLE integration using the key I have extracted with the method 2. When I switched my android app to China servers to connect the XMWXKG01LM button and then switched back I believe all keys where reset. Xiaomi BLE integration asked me the key and I used my Mi credentials to extract. I believe that then it extracted the key for the XMWXKG01LM button also.
That means that it worked without me having a Xiaomi gateway.
Does this make sense?