My dog, the dalmation looking beast in the picture, is just point crazy. Always has been, he points everything, and at times it is a little frustrating. However, when in grouse coverts he is fairly focused, except when it comes to deer pointing, or turkeys.
I can always pretty well tell when it is a turkey point. The dog is unsteady, looks back at me, not focused, relocates repeatedly. A flock of 10-20 turkeys puts out a lot of scent and they'll point them from a long distance.
You have beautiful animals. If they point a few unwanted critters, that's not a big deal. Stomp the mouse flat and reward the dog for his hunting skills.