Keep Sidecar enabled after restart, or re-enable it without seeing screen.

I have a Mac mini M4 Pro running Sequoia 15.6.1 and an old iPad (9th generation) running iPadOS 26.0.1.

When traveling, I would like to use the iPad via Sidecar as the only display. The Mac would have an HDMI dummy plug or a software virtual display as the "main" display, which would be mirrored to the iPad.

I can set this up using a separate monitor, then disconnect it and Sidecar continues to work fine. Unfortunately, after shut down and restart, the iPad won't automatically reconnect, even though the advanced settings "Push through the edge of a display to connect a nearby Mac or iPad" and "Automatically reconnect to any rearby Mac or iPad" are both on (and still show on after restart).

Is there a way to make the iPad reconnect automatically?

Now, to reconnect I must go to System Settings -> Displays, click the blue down-arrow and select "Mirror or extend to (my iPad)". How do I do this without seeing the screen? Try as I might, I can't find the keyboard sequence or shortcut for that down-arrow.

If there is none, surely there is a way to accomplish the same state change from the command line or by editing a file, but I have no idea how to find that.

I have a working solution using VoiceOver, but it requires many steps (focus on Dock, select System Settings, choose Display, etc.). Is there a good way to automate most of this?

Or, any other ideas on how to automatically restart Sidecar?

Thanks.