Hey Dorian O.,
Thanks for the question. I understand that you are receiving notifications on your Mac to approve iCloud keychain. Have you tried using your iCloud security code or your SMS-capable device instead?:
Frequently asked questions about iCloud Keychain - Apple Support
https://support.apple.com/en-us/HT204085
How do I set up iCloud Keychain on a new device without another device to approve from?
If you can't access any of your other devices that are using iCloud Keychain, you can still set up iCloud Keychain on another device if you have these items:
- Your iCloud Security Code
- The device that is using the SMS-capable phone number you provided when you first set up iCloud Keychain. A verification code is sent via SMS to this phone number. If you can't access this number, contact Apple Support, who can verify your identity, so that you can complete setup on your new device.
After you complete the setup, your iCloud Keychain will be pushed from the cloud to the new device.
Thanks,
Matt M.