UPDATE! Her name is Michaela.
We went to Walmart and to our delight, they had Liv wigs, sold separately, for $5.99. Auntie Berman kindly bought the Liv dolly a long blonde wig. We were all thrilled with the results.
Now if I could just think of the right name for her.
She will need a bigger hat, now. This one's a little too small now. I took the tiny rubber bands out of her hair.