Let us know, Wildflower. I use a lot of fresh ginger too and I haven't tried giving any to the girls. I do know they don't like other strong herbs from my garden. They avoid my lavender, basil, marigolds and geraniums. All of them have strong odors.
Just add some water to it again. When you make candied ginger, you usually boil the ginger for half an hour, and use the liquid that comes out for tea. Granted, we usually peel the skin off ginger before using it, so you might just compost it, won't hurt the birds.