How to reliably migrate emails from Mac Mail to Google mail?
I think that the main Issue is the Import process into Gmail which has several restrictions, f.e. Bandwidth-Limitations, and automated functions, f.e. automated filtering, labeling and sorting.

First of all, with 11GB of mail you need Patience!!!! 

Google Infrastructure has a bandwidth-Limitation on Traffic, see Google Support here for details. As mentioned there, 500MB IMAP Upload/per Day is not much and considering the amount you are planning to upload will result in quite some Migration Time.

Another point would be the rebuild and reindexing of the Apple Mail Mailboxes, see the Apple Support page here. Only the you can be (reasonably) sure that the Mailcount is correct.

Another Recommendation: While Migrating, make sure that the Account is not in use by other Clients!!!! There are several Issues in which multiple access to an IMAP Account can result in conflicting operations and filtering. 

Also, try not to repeat the same procedure to often - Google has a limitation there (see earlier mentioned Support page from Google)

That's all very general, but i think you should be aware of the limitations first before planning and executing a Migration of that scale. Letting your Client start something like that and trying to fix the Problems resulting from 'simply copying mails' does not work on that scale.