If you can see the email within the gmail web interface in the inbox (not a folder) then it isn't something Google is doing, it's how The Bat! is deciding not to download or show them.
Last time I saw something like this it was due to illegal characters in a folder where somebody had set a rule for the emails to go into automatically, the IMAP client then wouldn't download that folder due to a backslash I think.
As a test you could try another IMAP client like Thunderbird, just to see if it's appearing there.
Hang on...did you say it's your SENT items that aren't appearing?
If so that will most likely be due to mapping of the Sent / Trash IMAP folders within The Bat!