apart from any sync apps which service providers wont allow between different systems, you eiyhet set an alt recipient to deliver to both mail accounts or use a outlook clinet/server side rule for those accountz you arr working on.
Could be a number of things, verify autodiscover is working and set correctly.
Run the test from the client, i.e Outlook sys tray icon. Hold Control and right click Oulook icon>Test Email....>Uncheck Guest boxes, and test
It's your data they have, most providers that I have worked with are obliged to provide the data in a format that can be read and accessed by other systems.
I would escalate this and get them to provide your data.
maybe see it as an opportunity to give everykne email rather than them two.
If not O365 may suit you better although from what I remember yiu need a min of 5.