I have a gentle giant - Great Dane x. Similar set up to your house. We let them have blocked access at first (so maybe cardboard to cover the baby gates). When we saw she wasn't too interested, we gave gated access. We've had her 14 months now and she's started supervised visits to the bunny's room. One of the buns loves darting between her legs as she walks around!

I'd say look at getting a dog who is specifically good with small animals (we adopted our girl who had lived with kittens). Stay away from high prey drive dogs. And go at a timeline that suits your animalls. Maybe you need the cardboard a week, maybe you need it forever. Just make sure your bunnies are comfortable and your dog is not overly interested in the area before moving on.