Finally found the correct bluetooth configuration to let my bluetooth keyboard connect at login.

This allows me to use it on the login screen to enter my password in Manjaro Linux.

Edit the file /etc/bluetooth/main.conf and make sure the following options are set (simply search in the file for the configuration key, uncomment it and change value):

[Policy]

ReconnectAttempts=10
AutoEnable=true

These settings let me enter my password at the login screen without much hassle.