The red slime is an indicator of nutrients in an abnormal ratio... essentially bacterial activity. With zero nitrate, I can't see how chaeto is growing. What color is the chaeto? dark green, light green, washed out?
I would reduce the photo period on the refug slowly, use a GFO or other Phosphate absorbers. If you were dosing acropower before, I would continue to do that so as to provide proteins and slow down tissue recession.
don't get me wrong, having 1 ppm phosphate doesn't always means bad. I have maintained system with over 2 ppm phosphate with no ill effect, but also my nitrates were proportionally high (trust me, in the 40's).
Many of us in the hobby do use the clothes washer for the socks cleaning, but haven't seen a phosphate increase side effect to that. I can't argue that it cannot be. So changing that wouldn't hurt.