As someone who majored in animal disease in college, and a long-time parrot and cat keeper, I am pretty sure cats don't carry any airborne bacteria that will harm eggs. Cats get a bad rap they don't deserve when it comes to being a disease carrier I am surprised to still hear myths from people about cats getting into baby cribs at night and killing/infecting a baby just by breathing on them! I was told when I got married that I needed to get rid of my cats immediately because I should not have cats and children together because cats apparently walk around breathing diseases into the air! The bacteria cats have that is harmful to birds is in their saliva. They would have to bite/lick/mouth a bird for it to be a problem. The claws can also carry germs like this, but again, the cat would have to use its claws on the bird. (This is why cat scratches and bites are more prone to infection than others, they introduce this bacteria into the wound) They do not shed it off their bodies or in their breath in any significant way. I have always kept cats and uncaged birds together, interacting, and never had an issue in many years of doing so--other than one bird who would attack cats! I have owned many pet birds and none were ever confined to a cage very often, the cats always had access to them, and they often lounged around on chairs and windowsills together. The bacteria in cat saliva is believed to be an aid to them in killing their prey, as it can knock down a small animal very quickly if the cat itself does not kill the animal. So it is recommended not to ever let them interact without proper supervision and training. Even the dreaded toxoplasmosis carried by cats is not that common, and you get it from poor litterbox sanitation (i.e. fecal-oral contact, by not washing your hands between cleaning the box and handling food/touching your mouth. Yuck!), generally only from cats under a year old, and only ones that have been outdoors killing and eating wild animals. Older cats, and cats that get regularly dewormed generally are not carrying it. I would keep the cat away from young birds, of course, because what cat can resist tasting or pawing a fuzzy little chick? But she is not going to make them sick through the walls of the incubator and the shell of the egg just by hanging out around them.
The bigger issue would be with the cat knocking the incubator over, blocking vents, unplugging it, messing with any control knobs/switches, getting it open and playing with the eggs, that sort of thing. You know how cats are lol...mine wants to help me when I turn the eggs, it looks like great fun to her to roll those neat toys around! So your birds are not going to catch any horrible disease from her, but she should stay off anyway.
More ways to keep the cat down: Double sided tape. Stick it to the area around the incubator, and the lid. Cats don't like stepping on the sticky surface at all. Also, a spray bottle full of water, and when you catch her up there, spray her with it, preferably in the face. If that doesn't bother her, add a bit of vinegar to the water and it will. Or, a can/bottle full of gravel or pennies, something noisy. When you catch her up there, toss it in her direction, and the racket will annoy her and eventually she'll leave the thing alone to avoid the noise. (that one works best for cats that are easily startled by loud noises, and not at all for some cats. Mine finds that method amusing and tries to get me to throw the can)
