Indeed you should, it makes no sense to pass the cursor just to transform it afterwards.
As for the API I don't know, copying obmat should work no idea why is not in this case. But poke @Campbell Barton (campbellbarton) on blender.chat, he knows the matrix/copy API inside out way better than me.
Now what bothers me about this (although unrelated) is that when converting it should "delete" the original object.
@Dalai Felinto (dfelinto) I have talked with @Jeroen Bakker (jbakker) and he confirmed the right way to do the things is copying the rotation and scale because the matrix is recalculated using these values. I think we could apply this patch and make a different patch for removing the converted curve, as this is not related to this bug.
I just glanced at your replies, but some comments:
It is better to remove the object as a separate patch (sorry for mixing things up).
As for multiple curves follow what blender does for meshes and curves. (I won't be able to test this today I'm afraid, but feel free to commit to blender2.81 if you think you got it).