Of course you finish it! The quilt maker didn't piece the top just to piece it. Personally I would use a cotton batting and muslin backing in keeping with the time frame of the tops. Whether you MQ or HQ is up to you; as I do very little HQ because of arthritis I would MQ just so they would get done. What a find--good luck
Below is link to one of Eddie's vintage quilt tops MQ. As it's VQT # 5 there must be other quilts that he has posted.
http://www.quiltingboard.com/t-45978-1.htm