But it seems as if that password is burned into the phone forever and
that I cannot create a useable backup now or in the future without
having the old password.The only useful suggestion I have found is to Reset the phone, not
clearing data, which some posts suggest will lose the keychain and
Wifi passwords.
The password is stored in the iPhone (and iCloud and local backups of the iPhone) until the setting is changed or a settings reset. Unfortunately, a settings reset is the only mechanism that Apple provides to clear out the iOS backup password setting.
@aivar-paalberg made a comment on your question which is a good one, that if you use a Mac, you can look in keychain access to see if the backup password was saved back when you made it. I outline how to do that in this blog post trying to help people who have forgotten their iPhone backup password.
So if that’s the only way, is there a way to separately backup the keychain and Wifi passwords? Obviously I can’t rely on the normal backup system.
I believe iCloud Keychain and iCloud backups are your only alternative given your parameters. Even if we just stored the keychain from an iPhone backup, the iOS keychain is only backed up in the local backup if encrypted backups are enabled, and that keychain backup is encrypted with keys from the backup keybag. So it would still require the backup password to access.