I had exactly the same problem, I felt guilty building another run which was smaller than the big backyard, but they actually loved their own space and were really happy that the chickens couldnt get in.
I would really recommend building it!!
Is he the only drake?
I had a drake who really picked on my chickens.
I built their own pen and have had no problems since.
Perhaps if you can rehome him and get the duck another friend, female! lots of eggs and no dramas.