You have to rig some way - rope, weights, some way that Happy cannot get through the door when you're not there. If necessary, also put a bell rope on the door so you can instantly hear if she budges it even a bit. Wedge a chair under the doorknob if you have to - it's up to you to be smarter than she is, and block her from access to them unsupervised. And if you cannot trust her in the room with them when you're asleep - well, she'll have to not sleep in your room.

A dog with a strong prey drive is not going to "learn" it away. There is just no easy way to contradict her very nature.

My very first best dog in the whole wide world had an incredible prey drive - if it wasn't human or canine, and wasn't fast enough to get away by flying or climbing a tree, it was dead. There were no woodchucks in our neighborhood for years after she died, she had cleared them out of a wide radius - which all the gardeners loved. But all the cats in the neighborhood were quick and knew at any given moment where the nearest tree was, for their own safety. I loved my Sheba, and still do to this day, but we had no other animals when we had her, for that reason.