I've been apprised that the DNS entries above won't work because you can't CNAME a naked domain to a subdomain. Is that correct?
Yes, as stated in comments, you can't create a CNAME record on the domain apex, ie. example.com. Trying to do so can cause problems later.
You would typically do this the other way round. For example:
example.com. A 14400 ...
The only way it would hurt SEO is if the new domain had previously been registered and used in an abusive or black hat manner. Just registering a new domain, with no history, and 301 redirecting it will be fine. (But do make sure all the redirects are 301s - permanent - to ensure search engines continue to index the existing domain, and not the new one.)
If you are using CloudFlare purge cache and switch to development mode.
Some cache plugins can do redirects as well. Better empty the cache directory.
Theme plugins might have some weird issues as well.
Making a backup and deleting plugin directories one by one can help to find bad apples easily. (In case the logs aren't helpful.)
Maybe this saves some ...