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