Sounds like with it being a large customer of ours it may be best to ask their IT department to whitelist our DKIM as Mr. Whiteman suggests here:
We had a similar issue recently while doing a targeted account-based campaign to a particular company (about 300 names). Is it possible to do something different in the sending of the email so that we don't hit a spam filter? I haven't been doing anything other than putting them into a send list. I really need to be able to send emails to one account in a targeted fashion - do I have to actually do something with the send lists to make them smaller? Any help would be greatly appreciated!
If you send all at once, then you're leaving yourself at the mercy of the recipient's mailserver — and, even worse, you're compounding the problem by lowering the ongoing reputation of your domain and/or IP on their side.
You'll need to send to small lists, spaced out much more than you might expect. Start with 50 (yes, 50) per day. Work up or down from there. If you can, find someone in IT who can tell you the limit (this can be very difficult due to internal politics).