At the moment, I don't know what is the most appropriate way to export files from Brainstorm to MNE-Python. It was part of my summer work plans to write functions to create MNE-python objects from Brainstorm in order to call some of the MNE functions we don't have in Brainstorm, but I have not started it yet. Maybe by the end of August I will have something available in this direction.
If you are interested in working on this, on your own or together with me (but a bit later), the idea is first to create conversion functions, similarly to what we've done with FieldTrip (functions out_fieldtrip_...m and in_..._fieldtrip.m).
The Brainstorm data structures are documented in the sections "On the hard drive" of the various introduction tutorials, summary available here:
Examples of how to call Python and MNE-Python functions frmo Matlab are available in brainstorm3/python.
@Alexandre Any additional suggestion?