Hello @Rising Flight,
I can tell you cross-premises permissions are one of the more difficult and challenging things to get working in an O365 hybrid deployment. So I wish you the best on your journey to find this one out.
Here are a list of things to check
1. First, check the remote mailbox object on-prem. Is it created correctly, is the target delivery address correct? This is the first step in ensuring everything moves on correctly.
2. Check the connectors. I assume you set them up using the Hybrid Configuration Wizard. Think of running through it again. They need to be setup and working properly, allowing communication to flow through, and for that communication to be seen as internal.
3. Send on behalf of I have ALWAYS had problems with. It works if setup before, but once migrated its a pain. Specifically delegates are a bear to deal with - https://support.microsoft.com/en-us/help/4039613/delegate-cannot-send-on-behalf-of-after-migration-t...
If you have a fully updated AADC this may no longer be a problem, but I have had this issue in the past.
With that said you have 2016, so it SHOULD work - https://docs.microsoft.com/en-us/exchange/hybrid-deployment/set-up-delegated-mailbox-permissions
3. Did the permission get written properly? You should be able to see the permission on-prem. If you are trying delegates did that propagate through.
4. Do other permissions work? If you temporarily grant full access, or send as, do either of those permissions work?
5. Are you sure you are sending mail properly with send on behalf? - https://kb.intermedia.net/Article/2165
Start there, I would think one of those steps will lead to the issue, which at first glance im guessing is the delegates not writing back. Trying to add that manually, or trying another permissions should give you an idea.
Adam
