Thanks for posting this question. I too have had problems with Meshmixer when I want to perform very precise cuts or moves. For most instances Meshmixer is great when cuts can be "eyeballed," or made visually. But the plane cut tool doesn't seem to have a way to be really precise about it. In these instances, I've used Blender. Basically I will make a "plane" by starting with a cube and making the dimensions something like 1000 x 1000 x 0.1 mm. Technically this isn't a plane since it has volume, but if it is 1/10 of a mm thick for practical purposes that is a plane. You can then set the exact XYZ coordinates and the XYZ rotation of the cut plane in the object parameter pane. Then, do a boolean difference to subtract the intersecting volume between the plane and my target object, effectively making a 0.1 mm thick cut through it.
Blender is scriptable, and I've had instances where I have to do multiple identical cuts on multiple objects, and I've written a python script to automate moving the cutting object and performing the cuts.
Sorry this isn't specifically a solution for Meshmixer, but hopefully it will help you.