I would suggest you contact the Exchange administrator and do the cutover migration with him.
After cutover migration, every user will have two mailboxes, one in Exchange on-premises(User@test .com), another in Office 365(User@test .onmicrosoft.com). Those two mailbox are in different organization which don't affect each other. You can configure both of them in the same Outlook profile.
The Cutover migration will also help you migrate mailbox data from old mailbox to new mailbox. For more detail information about this migration method, you can have a look about this article: Migrate email using the Exchange cutover method
The migration action is an Exchange admin job, but the Exchange server is hosted on your organization. So, I think you need to cooperate with the other party’s administrator to migrate mailboxes.
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