So the benefit of having many domains to play with is that I get to test things out before I put them on a live site.
I was able to successfully transfer a multisite WordPress installation to @Dreamhost. Here’s what I did.

1. Use 1-click installer to create base site @Dreamhost.
2. Make site a multisite installation.
3. Use SFTP to copy existing plugins/themes.
4. Use WP Migrate DB Pro to push existing database to new database.
5. Create mirrored subdomain in @Dreamhost panel for subdomains.
6. Remember to check Remove WWW from the main domain.

Shout out to @Dreamhost support for pushing me in the right direction.
Next up, tackling IMAP email transfer.