For your Suruga sucker, I would graft it with the Suruga now. It may take or if not wait until next spring when leaves grow again. Jt would be good insurance in case the main tree has health issues later, then you can just cut off the sick trunk.
For the native mulberry, I would keep the sucker to use for rootstock to graft some good named varieties like the Australian Green or Red Himalayan. Send me a message if you want to buy a few fresh fruiting scions.