I'm having the same problem as described here:
http://www.mozilla-enigmail.org/forum/viewtopic.php?f=4&t=425

The prognosis wasn't promising:

Re: Enigmail & Exchange 2007
by patrick » 11th Jun 2008 09:18

The problem here is that the 1st message is wrapped inside another MIME part.
Unfortunately, this makes Thunderbird not detect the message content type (since it's
now a sub-part of the message), such that Enigmail is not invoked. I have tried several
workarounds for this, but it's currently an unsolvable issue.

.
I'm using MT 17.0.4 and Enigmail version 1.5.1 (20130205-0013).

Is this still an unsolvable issue?

Is it possible to convince TB to give all messages to Enigmail for possible decryption?

I don't think that this is fix-able. TB simply ignores the PGP/MIME subpart.