I think moving your shoji screen to her side is a very good idea. To soften the sound of her voice, DH could wear earplugs. Otherwise, he could say, "Please don't distract me during my workouts - I could injure myself. My evil twin would say, "and if I'm startled, I could lose control of a weight and it would go flying (in her direction)."
Do you hear her when she calls out to your DH? Perhaps when you hear her, you could go outside and just look at her like she's the most pitiful thing on earth. Then shake your head, kiss your DH, and go back inside.