Sync data between Exchange Office 365 and On-premise Exchange Server 2016

I’m looking for some advises to know if it is factible to install an Exchange Server 2016 on-premise and if there is a proper way to implement it with a hybrid solution.

First of all, our customer is an office from a multinational company and it has a responsible for email accounts globally. Recenently, he adopted Office 365 with Exchange in cloud with 50GB mailbox per account.

The problem is that our customer receives to many emails internally and externally on each account. After uploading their PST files from the old email system to Exchange and after 2 years of usage, they reached those 50GB. Currently, we are saving old emails into a backup PST file that they can use to find and filter old emails, but they doesn’t like that. They prefer having everything in the same folder. Also, they refuse to delete old emails, because they lack on a CRM system and they need to store the email information from customers.

Another problem is that we do not have access to manage those email acounts, because as I mentioned, those accounts are managed globaly by the company responsible. What we were thinking, and we do not know if it is possible to achieve, is to provide a hybrid solution “linking” accounts. For example:

Exchange Office 365

User: user@contoso.com

Exchange On-Premise

User: user@contoso.com

A user sends an email to user@contoso.com, Office 365 receives it and it is automatically downloaded to the On-Premise server with the same account. When the process is done, the user, that is connected to the On-Premise server, receives the email and it is removed from Office 365 (like a POP3). When user wants to send an email, the On-Premise server sends a request to the Exchange Office 365 to send an email to the desired account.

More or less is what we want to achieve. Is it possible or is the hybrid solution only working for staged migrations?