I would offer the stud owner what a reasonable stud fee is in your area. She can either walk away with nothing from this breeding or she can make a couple hundred with the agreement that she gives the studs info.
But in all reality you would have to give up a few hundred which is probably the loss you will take having to sell your puppies unregistered.
To make this work both sides would have to compromise some, I would learn my lesson from this, walk away and prevent it from happening in the future.
Congrats on the babies