We paint small round stones with red paint here, but those are near our berry bushes, so they look bit of berry and that makes birds then learn to not bother with the berries. I don't know if it would work with cat food as they probably don't associate food and painted rock to be the same.
Putting stones to cat food (enough large cats are not eating them and same color as cat food) might work, cats have lips, and tongue that they can easily tell difference between rock and food, but it certainly does not sound very great idea to put stones to their food.
If it is group of birds, taking down few would teach group to stay away, they learn quite well, but we have Eurasian Magpie here, it is able to learn too well, knowing when one is hunting them and when it is safe to steal food.
Birds rarely use cat door, imo. So making feeder station which has cat door might be one way, but not all cats learn to use cat door by themselves and with ferals at least it is difficult to teach them as they would need to see how to operate the door.
Bastfriend, lot of birds eat insects and owls, hawks etc, even hunt actively, some are danger to cats too, crows and alike eat mostly already dead ones, but for example some woodpeckers can eat small bird chicks and eggs, so definitely some birds are carnivorous.
Nets can be used of course, but they tend to kill birds and if net is wrong type even cat can get tangled to it, but even at best nets tend to then trap birds into area isolated with nets, so they are usually more problem than a solution, but many people use nets to keep birds away from berry bushes.
So, does it come to only possible solution, using old clothes and hay to build human resembling figure that moves at wind? To improve effect one should stalk birds and shoo them away with broom, then put broom to that figure that is used to scare birds, cat's perhaps learn quickly to ignore such figure?
I have scared magpies away with air rifle, shooting near them to ground, one must be careful not to hit birds, also it is not needed to hit very close, several feet away works too, they get scared from pellet hitting the gravel, of course this requires one to spend time to stalk them, also it works mostly on gravel, shooting the lawn is not so great effect. Blanks shooting gun might also scare them, but I find they learn too well that it is that person doing the sound, best scare is such that they don't know where it came from but was scary.
Oh and one could always put bird feeder some distance away, but it too is extra cost, however there might be food better tasting for them at feeder, so if one does not like from making them go away, but keeping enemy near approach, of course then one might find enemy being a friend