Sorry to report their answer:
At this moment there is no counterpart for StructuralConnectionHandlerType inside Advance API and there is no method to edit parameters for StructuralConnectionHandlerType. We filed a request for this for future versions, REVIT-152546.
At least you know where you are with this now.
That said, you can try to access the standard generic Revit element parameters and see what you can do with them, just like you can for any element.