An issue with MiaB is long TTLs, which makes this sort of thing somewhat more difficult. The best I could offer is run imapsync to add the new account messages, make the necessary DNS changes, keeping both server DNSSEC keys on the domains being transferred (this is permitted under RFC for exactly this purpose), probably have to manually make any custom DNS entries, and run imapsync somewhat regularly until finally everything is using the new server.
I’m not sure how to deal with active user accounts, other than have them create two separate accounts on their clients. This is probably a quagmire if you have a large number of users.
Or maybe there is a way to create a second mx record, since MiaB default is priority 10, you could make a priority 5 mx record that is for the new server, and then update SPF records so sending passes with the new server IP address.
Honestly, just kind of brainstorming here. Was hoping someone else would be able to suggest something for you.