Understand.
First Aqua and Type Mail operate on entirely different principles - not that it should make much difference to you as a user, but

Does the Exchange account have push enabled?
Could you as a test turn that off? ( long press on its on the app's main screen -> options and folders -> account options ) to see if it makes a difference?
Also feel free to capture the issue in the app's debug log (linked below) with "raw session data" turned *on* in debug settings?
With the log enabled, please refresh the account or just inbox, maybe 2-3 times, waiting to finish each time. Then send the log file to support / at / aqua-mail / dot com.
Thanks!