The patch looks good, but I don't think this is all that is needed to port to 2.8.

I would expect that for export you need to get the animation from the evaluated datablocks in the depsgraph, so that it includes all the f-curves, constraints, etc.

hm ok, will need a better file to test this then.

I guess just add a constraint that has some visible influence, then see if it is taken into account for export.

I tried it with this file:

It exports the animation with the constraint applied.

One thing I noticed is that the default axis conversion of the exporter and importer don't match up. So when I reimport the .bvh file, the armature is rotated.
Should that be fixed or is there a reason for it?

[EDIT:] oops, wanted to send the .blend file:

It seems the bone matrices get flushed back to the original object so it ends up working, a special case like the object matrices. So it should work ok for this exporter then.

Definitely seems like a good idea to make import/export do consistent axis conversion.