I'm sure you've seen this:
https://help.astropad.com/article/218-how-do-i-use-luna-display-in-headless-mode
It notes that to set up a Luna Display to use an iPad as the only display still requires a 'real' display be connected to the Mini for the initial set up. After that you are ok just using the iPad as a display.
I have a Luna Display and used it with my old iPad Air. I'm not using it now, as the Air has been retired (battery issues, very sad; a new iPad is in my future). But I was happy with it when I was using it. When I first set up my Mini, I was having issues with the eGPU (since sorted). And I used the 'headless' set up to get my Mini to start up, after which the eGPU would then bring my 'real' displays online.
And I'm not sure you understood the idea behind the adaptors mentioned above. I use an HDMI adaptor in my Mini. Not for screen sharing, but rather to trick the Mini into thinking that there is a display directly connected to the Mini. This is what allows the Mini to boot up and enable the eGPU. In your case it would enable the use of Sidecar, as the Mini would think there is a display attached.
Good luck, and I hope this helps.