I am trying the Facebook Customer Chat plugin on my website. This plugin sends messages directly to your Facebook page. Now I want to get notifications on incoming messages right away and be able to reply using the webapp https://www.messenger.com
However, messenger.com only shows messages from my private account and I cannot find a way of how to switch to my Facebook page.
I do not want to use: https://www.facebook.com/MYSITE/inbox
Is there any way of how to add the page to messenger.com?
PS: For my mobile I found this Android App: Facebook Pages Manager. But I'd like to be able to reply on messenger.com because it is way faster.