Yes, you can provide your ID and request a copy.
Here's the problem as I see it. Your daughter has been settled, under the care of your mother, for 1/3 of her life. Is it in her best interest to be uprooted from that? You are in a shelter and you're not working - your Mom might be able to make a case for custody if she fights.
Given that DFC was involved, this is something you may need an attorney to help with.