I’m so happy to see her doing what Arabian mares do best! And I’m very happy to hear that you were able to get her registration transferred.
Any registered foal I have purchased was already registered. Sounds like the breeder has been a bit lax and it is great timing for them and an expense they don’t have to fork out now.
I don’t know the breed you are referencing but it usually isn’t terribly expensive to get a horse registered ( IME) but when you add the joining fee, registration fee and then the transfer fee it will just depend on the breed registry prices.
I personally think the breeder should pay the fee but I wouldn’t let a good one get away over the registration cost if it is doable. Just get the completed , signed paper work and everything else in hand before you pay and leave.
ETA: Once again I posted before reading the whole thread and see the OP has passed on the horse! No matter as I will leave my opinion on the subject anyways.