I'm not a member of the bar in MS, so this is
NOT legal advice. Just my opinion, which is that there may be a contract because there was a promise and performance, although even that is questionable since she lied about the registration.
Even if there is a contract, it seems to be partial performance (No papers) so only partial payment is required. It seems like a check for the stud (which was performed) marked Paid in Full would meet your responsibility under the contract. Giving her the puppy would be payment for full performance on her part, which has not taken place.
Part payment for actual services provided would discharge your obligation, so you can feel ok about doing it. Her only argument would be inadequate time to perform (the papers), but that's a matter of opinion and you are not being unreasonable in your expectations.
If it were determined that there was no contract because she lied about a crucial term, you would owe her nothing. But I see no reason why you shouldn't pay her the minimum for an unregistered stud, if that helps you to put this behind you. Do what you feel is best for you and the pup. You'll get lots of support here!
I hope this helps you lay this matter to rest.
Repeat:
This is NOT to be construed as legal advice.
Can you tell I'm trained as a lawyer?
Good luck from me and Samantha,
Vivian
Oh, do keep the pup.