Marilyn covered most of that. Johanna and I were just adding filler on Hide's behavior.
Anyway, the problem got my curiosity up so I did a little searching and testing myself.
Everything I read and tested suggests that Gmail does not save the BCC info to the Sent Mail folder when the email client uses IMAP. It apparently saves your sent emails as if they've been sent, probably by sending them to the Sent Mail folder, and since you don't want the BCC info in the actual sent email (or the recipient could dig into the header info and see it) that information is not there.
While I did not dig terribly deep, the consensus seems to be that you can solve this for sending emails via the Mail app on the iPad. On a computer email client you can choose to send the Sent email to a local folder instead of Google's Sent Mail folder, bypassing SMTP/IMAP which appears to be at the core of the problem. iOS doesn't offer a local folder option for sent emails in IMAP accounts.
Kind of solutions:
The BCC info does get saved in the Sent Folder when using the webmail site. I guess because webmail sites do not normally use IMAP or SMTP. I'm not sure what they use, to be honest, but since most of them have direct access to the email servers POP, IMAP, and Exchange are unlikely to be used.
So, you could try using Gmail in Safari, or use the Gmail app from the App Store. That also seems to retain the BCC info.
Another, outside, possibility is setting Gmail up as an Exchange account instead of IMAP, but you need to have one of Google's paid accounts to do this, and I can't guarantee it would fix the problem.
I don't know if it is possible for Apple to fix this. It appears to be entirely on the Google side of things and about IMAP. You can however send Apple feedback, just in case. You should send Google feedback as well. I doubt a fix is trivial, since it sounds like it's a problem with IMAP, or at least Google's implementation of it, which is not very standard.
Apple - iPad - Feedback