You can get failure notifications on the Chrome browser using the Post SMTP Chrome extension by enabling push to Chrome extension and providing Chrome extension UID.
An extension is an extra option to the other notifications options, what I mean is you can enable it as an extra option not instead of what exists today.
The Chrome extension can be downloaded here: Download extension
For now, you need to log in with a Gmail account to create a UID (unique identifier), so the plugin can communicate with the extension.
For the UID you see in the extension just paste it into the settings under the notification tab.