How to set up sidecar without giving Apple your account password?

If you equate giving apple your password to typing the admin credentials, then you can’t do this without a privilege escalation / security vulnerability. Those usually pay $25,000 to $250,000 so if anyone has one - get your money from apple before posting here how to hack your mac.

• https://developer.apple.com/security-bounty/payouts/

If you just need us to confirm you aren’t sending your password off device in the prompt you showed, you are set. Use the admin password and understand that Apple doesn’t send your password off the device, it entangles a local key with a downloaded public key and then stores the results in your keychain.

Apple already has your iCloud password so it doesn’t need to record / steal / transmit your local password for sidecar to work. I understand you would love to have someone show you the way, but as I read it, your intentions are good but your threat model analysis is faulty here.