Okay, you have got me curious.
22.214.171.124 is def google.
This line here though
Received: by 10.25.17.25 with HTTP; Tue, 7 Jul 2015 09:43:37 -0700 (PDT)
That range, that seems like a apple device. I am not convinced they didn't use an email client. but hard to argue with with HTTP; Customers don't always..... tell the..... admi...hmmm... sometimes they get confused with our questions and their answers do not meet our criteria :P
What did your delivery log say? a bounce email was written.. so there are a few lines, probably nothing helpful.
I would set up a test google account. Connect with an email client. Send to a test account that you create on your customers domain. Send from google web mail, and one from google/outlook (email client)
If at all possible, have this other company, that has been pretty helpful send to your test account on your customers domain.
Your issue really has me curious, Let me know how it turns out.
Remember kids, every time a spam message gets blocked, a nerd gets their glasses. spamhurts/July 15