Bevins
In the worst scenario, they may toss them, although it is more likely that they would just abandon them.
But it depends on the bird's temperament, some are easy do deal with, and some are more scared / nervous.
Personally, i don't like to handle the nest when they are breeding, the less we interfere the better.
Just asking, if they are brooding in the bird feeder, could the incubation and all the rest go well?
If i were you, i would wait until the babies are about 7 days old, it is less likely that they would abandon chiks this old them the eggs, or small chicks.
If you can't wait, so do it carefully, and when both birds are out of the nest, so they should be less disturbed this way.