The problem is on this line:
https://developer.blender.org/diffusion/BA/browse/master/io_scene_obj/export_obj.py$376
bm.to_mesh considers the shapekey offset between the bmesh and the evaluated
object, thus changing the object already transformed by modifiers.
I'm not sure if me->runtime.is_original indicates that the mesh is always the
one not changed by modifiers, but if that is the case, it seems to be a viable
solution.
Ref. T76715