v4.823 should now keep track of the navigation stack
Maybe macOS doesn't show the BTT menubar but still has it active because the BTT process has keyboard focus when the searchable list is visible.
It looks like that. When I use System Settings to map the BTT menu Item open live view to ^+L live view is not opened anymore..
v4.823 should now keep track of the navigation stack
It does! 
Thank you!
Also, in this version CMD+L does move the focus to the input field, even if I do not remap the BTT menu item's key sortcut in system settings.
personally I like the search field highlighted for the main menu but not for submenus
When I enter the submenu the first item (Go back) is selected. When I start typing the first menu whose name starts with the first letter that I type is selected. However, when there is more than one item starting with this letter I need to go back to the input field using CDM+L or navigate using arrow keys. In case there are many submenus, it would be nice to first filter submenus using the typed input. Thus, an option for the focus behavior will be great (possibly also as option for the simple JSON format)!