When this was reported in the past and the BTMail email team investigated all they could do was establish which device triggered the deletion of the emails and on those occasions it was an Apple device that had triggered it.
Once the emails are deleted from the BT servers they can not be recovered.
I would suggest that you use a process of elimination to try and establish which device is causing it.
For example stop your Apple device's email app from checking for email or you using it to access your emails.
You can do this by just removing your password from the app.
You will need to use a web browser to access your emails. If you find that the problem is still there it would appear to be a BT issue. If the problem is not there it would appear to be an Apple issue or you could try using a different email app to see if the issue is still there.
If you dcide to do that make sure your Apple email app is not accessing your email account as well.
