The status bar icon has to do with how the app stays in memory, avoiding being killed, while checking mail or working in push mode.
This (the icon) is "Android by the book", and is the Googlers' decision.
If you turn the icon off, the app will utilize its own means to avoid getting kicked out of memory (which will stop push mail).
So - do try to turn it off, see how it works for you.
One more thing:
It seems that every single app that reads notifications from other apps has the error of confusing "ongoing" and "event" notifications. Sad to see that Pebble's is among them.
There are two ways to check for ongoing notifications:
1 - Not specific for my app:
Notification.FLAG_ONGOING_EVENT
http://developer.android.com/reference/android/app/Notification.html#FLAG_ONGOING_EVENT2 - Specific to AquaMail:
The notification's Intent has an extra called "org.kman.AquaMail.IGNORE_NOTIFICATION".
Maybe you can relay the above info to Pebble support.