Well, a friend of mine does wildlife rehab, and it is not a bad idea at all to pick up injured wildlife as long as you keep yourself safe and keep the wild animal away from your pets, and you get a hold of a wildlife rehabber once you get the wild animal secure in a safe area. I would NEVER ever walk away from an injured animal, and at the very least I would at least put in a phone call to the wildlife rehabber if nothing else, it is cruel to just leave them. You will not get in trouble if you pick up an unjured animal and take it to the wildlife rehab place.
You sure did all you could to save that bunny, that was good of you to do. I sure can't understand why your mother wouldn't at least take the poor thing to the vet to be euthanized tho. You did good tho!