Time Machine Backups Password Keychain Location

Backstory: I've been dealing with a corrupted/failing hard drive issue on my MacBook Pro where I'm unable to boot up normally, but I am able to boot into recovery mode and transfer some files to an external disk via Terminal. I'm not asking for any advice on that issue in this question. I want to try restoring from my Time Machine backup, but I've somehow lost the backup password I had written down. I know the password has to be stored in a keychain because I've been able to do regular backups to my Time Machine hard disk whenever I plug it in without typing in a password.

I've been trying to locate the keychain file where this password is stored so I can move the keychain to another Mac, unlock it, and view the backup password. I've looked at all of the keychains in /Users/<username>/Library/Keychains/ and /Library/Keychains/ (most notably /Users/<username>/Library/Keychains/login.keychain-db and /Library/Keychains/System.keychain) and found all of my other passwords for internet accounts, WiFi networks, etc., but I haven't been able to locate the password for the Time Machine backup. Where else could this password be stored? And what would it be called?