Hi,
I think this is a topic for the Better Messages support forum, but you can try the following:
Edit the specific email to remove the part which includes the message text.
So, go to …
WP Admin ->Emails ->All Emails
Identify the specific email which gets sent (“You have unread messages”)
Edit it, delete the {{{messages.html}}} token, click Update, all done.
Hope that works for you.