Not sure what the lock screen has to do with AquaMail. I push the counts to Apex, and where they're used is up to them.
You might want to consult the developers of Apex, but as far as I know, the lock screen is a separate piece of "real estate", and can only be accessed in a documented way since Android 4.2.
So if you have 4.2+, consider installing DashClock widget (free in Google Play), and grabbing a recent development build of AquaMail from this forum (support added a while ago, but not released to Play yet).
For earlier Android versions, consider Nova Launcher + Tesla Unread, which use various undocumented hacks to add themselves to the lock screen. My app has native support for this combo, too.