Is she paying for the wedding? When a parent or other relative is covering the costs there are usually strings attached.
I would look for a compromise where the band she wants plays during cocktails or during dinner or for part of the dancing portion of the reception.
If your fiance can’t seem to talk his mother into some kind of compromise, it’s up to both of you to decide if you:
A. Need her money to cover the wedding costs and have to do what she asks
B. Can ignore her requests and hire who you want
C. Go ahead and hire her band as well as the band you want and have the two coordinate who will play when at the wedding
A wedding shouldn’t cause World War III but it’s common for individuals to disagree on the details. If your future mother-in-law won’t compromise then I vote for C and if money’s the issue, hire a less expensive DJ of your choice to alternate with the band of hers.