Logitech G613 Bluetooth keyboard stops working after first connection on 2019 Macbook Pro

My G613 Logitech Keyboard connects to my MBP correctly at first. The MacBook ask for a pin code, the keyboard is working and it works without any problems.

As soon as I power off the keyboard just one time, or put the MacBook to sleep, the keyboard stops working. However, it still shows as connected in the Bluetooth device list.

Some informations :

• Keyboard works perfectly on another PC
• Macbook is on Catarina, last available update.
• Catarina is an update, not preinstalled
• Keyboard worked before, can't remember if on catalina or not
• Tried a SMC reset
• Tried reinit Bluetooth module (shit+option on Bluetooth icon > Debug)

I faced exact same problem. This worked for me :

press and hold the Bluetooth button in the g613 keyboard until it starts blinking faster.

On Mac, do the BlueTooth connection as explained by @Vadim Flaks, once connected you can remove the old G613 from your BlueTooth settings ( which is now disconnected )

I wish I knew of a way to make this problem go away completely, but the best I can give you is a workaround:

1. in the Bluetooth menu hit the "X" button to remove G613 from the list of connected devices
2. hit the connect to "Bluetooth" button on the keyboard again to make G613 discoverable.
3. when the keyboard shows up in the bluetooth menu hit "Connect" and punch in the PIN as asked
4. You're back in business until the next connection interrupt. Rinse and repeat.

This, along with the useless and annoying macro keys, really makes me wish I went with something else for my wireless keyboard.