If the parents are taking care of the eggs don't bother them. Reaching in there to remove a few white eggs, which the parents themselves will do, almost guarantees the parents eating the eggs. GT's have been raising spawns longer than we have, trust their instincts with this.
The fungus you see is secondary to the bacteria you don't see. The parents know viable eggs from infertile eggs better than we ever will. If you remain aware of the bacteria you will have a better hatchout with any species. Clean, fresh water is a primary concern, and water changes are all that you really want to do with a new pair. You want to go as slow as possible with this, no quick movements, sneaking the siphon in at a point furthest from the spawn. The same goes for adding water back in, no dumping buckets, a slow stream far from the spawn.