These days I'd shove the whole sodding pile into Office 365. Archiving and Exchange. Let MS deal with it.
End of the month does not really give you much time to do anything about this. Unless you can get a temporary contract extension for a few months. As for getting mails out as .eml, frankly that's a joke. Paying to get that data out, most cloud providers have this sting in the tail, not quite vendor locking but it has that effect.