Sorry for the loss of this little one.
When they are that small do not overfill them. It is so easy to get some in the airway and they strangle or inhale into the lungs which is fatal or will turn into pneumonia.

Feed moderate amounts often when this young. I use the small end of a leg banding tool and let the chicks basically like the food off of it. That way they swallow it instead of inhale it. I have never had luck with a pipette or syringe.
Do not mix too thick, a runny pudding at this age is best.
Are you using Kaytee handfeeding formula?
It is tough when they are this young. They usually are not fed by the parents for the first 24 hours after birth. they are substained by the yolk sac.
Very strange that the gouldians would be laying new eggs at the same time these eggs were hatching.......Usually they can not have clutches that close together.
Could these be someone elses eggs? Do you have more than 1 pair or other species in the cage?