Back up cannot be restored on another iPhone

If your iPhone 15 had the iOS 18.1 beta, then I don't believe there's any way to get it to restore on an earlier version of iOS. You'll have to put the beta on your iPhone 12 and then see if the backup will restore.

Regards.

One thing you could try is turn on iCloud and have all your Messages and Photos (and whatever else is possible) stored in iCloud. Then see if the iPhone 12 has everything that you need.

Another option: back up everything on the old iPhone to iCloud, and then try to restore from that iCloud backup. I suspect it will not work just like the backup on your Mac did not work, but it might be worth a try.

Another thing you can try: there is a software app called iMazing, https://imazing.com/ that claims to be able to copy virtually any content to/from iPhones. You could try using that. I can't vouch for it personally because I don't have it myself but I see it recommended from time to time. I think you can try a free trial to see how it works before purchasing it.

Personally, I would avoid putting the beta iOS on any more of your iPhones, that is how you got into this bind in the first place.

Also, see this, Apple's instructions: How to uninstall iOS or iPadOS beta software - Apple Support