OH, I don't worry about the dogs. My Schnauzer has caught a day old baby Quail that my DS dropped. And ever so gently set it on the floor at his feet. I nonchalantly picked it up, told him thank you and put the baby quail back in the brooder. DS learned not to let the babies walk his arm. And he is an adult and should have known better.
No, these are their babies to protect. They catch them in mid air, check out their houses with me, check the yard, chase off Hawks, their job just never ends and it was their choice to do so naturally. I didn't train them, I just show them by doing and then they copy me in their own way. I can trust either of my dogs to be alone with any of my birds at any age. Not the cat.
The cat does ignore them but is curious about them. But like you, I would not trust her alone in their bathroom with the lid off the brooder tubs. If I had birds in the house, there is no question in my mind she would want to play with them. She is strictly an indoor cat, but she likes to play with small things. Twisties, bottle caps and probably baby quail given the chance. Might not hurt them, but they would still die from fright.
I had to house one of my baby ducks for a few days and I just put it in the living room in a tub and she layed by the box for hours upon hours, the duckling ignored her, but she wanted a friend.
This is going to be a long hatch. I think I better order a new heater, thermostate whatever for the bator for next year, along with a few plastic tops to replace the wharped one I have. Tape works, but maybe not that great.