There is no permission called “contact_email”, there is only “email”.
That being said, you need a Page Token to post “as Page”. Use
/me/accounts to get Page Tokens for your Pages. The permission error most likely means that you don´t have the appropriate permissions to post to the Page with your Access Token. Make sure it is a Page you manage, and make sure the Access Token includes the
publish_pages permission. You can debug your Access token in the Debugger: https://developers.facebook.com/tools/debug/
If this error occur during WhatsApp API configuration, Call me for this solution.