Bed partner or not, it would still be more comfortable.

Oops, on the belly

I'm really wondering if I could sleep comfortably on my belly when I stuff my boobs in there.

I think it would squeeze the boobs, not really comfortable

Ladies... sleeping on your stomach an issue? Solved:
http://comfybreasts.com
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I'm not affiliated with, or profit from them, FYI. I post this because if it can make your life better and/or happier by a small measure, perhaps your partner's may also be as well.
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Laundry day an issue? Boom:
http://brababy.com
Alternatively, I have also heard that you can use a salad spinner to "hand" wash them.