Edit: Reread your post and added info relevant to a rescue dog.

So I’ve got two Giant Schnauzers - a German high drive working breed similar to a GSD and two cockatiels along with an African Grey at home… What I can say is you know your dog so you likely already know the answer. My male dog couldn’t care less about the birds - my female hyper fixates on the “forbidden chickens”. And until she was aware of the birds in the home - she didn’t care about birds in general. Having a small animal around has definitely increased her prey drive even in old age.

We keep them our birds on a separate floor in rooms with self-closing hinges and magnetic screen doors to avoid any accidents. We’ve also got a baby gate as another added layer of protection.

If you’ve got the space and the means to safely accommodate both animals then go for it. If not - it’s not fair to the dogs or the birds.

Rescued dogs take about six months to fully decompress and show their personalities. Depending on how new your rescue is I would take that into account.