Theres no reason or benefit to changing which account “owns” the extension.
If you need you extension to be allow listed, use the allow list fields as needed, which you would need to do if the extension is owned by your “Streaming” account instead of “personal” account
I’m assuming you are using a different name to the one on the personal account?
If so sounds like a recurrance of Developer Console - Can't add new application · Issue #634 · twitchdev/issues · GitHub which is unfortunate