I wouldn't take out the quilting...I would have the LAQ do it. She messed it up, and as others said, she should have noticed it when she started quilting and adjusted the tension. Would not trust her/him to redo the quilting, only to take it out. I hadn't thought about putting another backing on it. I bet the holes will not be there after washing.
And, yes, this should be her/his top priority since you plan to give it as a Christmas gift.
Kathy