Current transformer with additional multi-turn winding for test purposes
Do I need to connect the test winding to the DC shift like the sense
  winding? Or do I need to connect it to ground instead?


You can connect it directly to ground because, with your drive circuit using a series capacitor, it fulfils the AC requirements when driving a winding of any transformer. But, you could connect it to any DC voltage within reason but there is no advantage so, connect it to ground is my advice.