Can I somehow save display configurations and switch between them with a keyboard shortcut?

You can bind scripts to shortcuts in systemsettings shortcuts.

Then you need a script to switch between the state.

You can use kscreen-doctor (command line) to enable/disable screen or request the state of your screens.
kscreen-doctor --help gives you example how to enable or disable a screen. You will need to find the screens name kscreen-doctor -o and available settings.
You can use kscreen-doctor -o | head -n1 | cut -f 4 -d ' ' to determine if your first screen listed is enabled.
Then it is a matter of if/else.
It might be a bit too technical.

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