I'm not sure what those berries are. The mulberry trees around here have a reddish purple berry. I just did a google search and saw that there is a white variety as well.
Cassie gets seeds daily. Her vet also said that the seeds are given in moderation. She gets mixed seeds with pumpkin but she doesn't eat sunflower seeds. The vet said that 1 to 2 tablespoons per day max. She also doesn't get the seeds until suppertime. I would ask your new vet what they consider moderation to be in a specific measurement. 1/4 cup per day seems to be an awful lot of sunflower seeds.