Error while using an iPad as a second screen -

I believe you need to be using your home Wifi network and not Bluetooth, at all!

Use your iPad as a second display for your Mac with Sidecar - Apple Support

Best of Luck to You!

To add to what MichelPM has provided, I can say that it works quite well with my iPad Pro 11, and my MacBook Air 2019. It has really improved my usage of Zoom, since I can put the Zoom controls on my iPad and the work I wish to share with my students on the MacBook Air and share that screen, while also using the camera and mic. I can put the AirPlay icon in the Menu Bar and connect the iPad quickly. It uses my home wi-fi, not Bluetooth.

Are you using a VPN client to access the internal network of an affiliated organization? If so, try what you want here again but with the VPN client disconnected. See if what you want then works. This as some VPN clients will block local network access, as part of a VPN security policy configured by the folks managing the VPN client. Cisco AnyConnect has been reported to block Sidecar, for instance, amd I’d suspect other VPN clients could be similarly configured.

To address the fact that you are not seeing an AirPlay icon on the Mac, do you have it activated? If you go to System Preferences>Display, do you see the AirPlay information? I also have the AirPlay icon in my top menu. Did you try all of the steps in the link that MichelPM provided?

Please post the EtreCheck data. Download that app, enable full device access for that app, and then open a new reply here and then press the button that looks like a printed page to get a text input box big enough to paste the hardware and software configuration report here.