Oh, the classic "I don't have permission" error from Meta. Shocking. Next you'll tell me the sky is blue and Business Suite is still a dumpster fire.
Here's the actual fix - but brace yourself, it's Meta's official "we know this is broken" page: [link to docs]. Basically you need to jump through a hoop where your WABA is verified and the person creating templates has admin or TPM role in the Business Manager. If you've already done that and it still throws the error, congrats - you've officially joined the club.
i've had this pop up after a perfectly innocent template edit. Support took four days to come back with "try again" and surprise, it worked. so yeah, sometimes it's just Meta being Meta. Try the link, then try screaming into the void. One of those usually helps