I have my new pup Peanuts and 2 cats. My one cat does go upstairs but I do keep my dog and 1 cat out. I supervise while the birds are flying especially my pup because he is reactive to them flying around. He does listen when I tell him to "leave them".
Pic of my new pup Peanuts and him with my 20yo cat Sammy. I got him from the pound. He had mange, kennel cough, full of flea dirt, and every worm possible. He also only weighed 4.5#s. He doubled his weight after living one month with me. He now weighs 9.5#s.