This is still partly eyeball adjustment, but likely to be at pixel resolution. These steps imply a horizontal plane placed with reference to the zero alignment of both the object and the viewport.
For my test:
I loaded the wabbit and added a plane (Meshmixer, Plane, Accept)
Edit, Align, Accept to place the plane on the origin.
Edit, Transform, Translate Y, 4mm, Accept to move the “reference plane” to the desired height. You may have to use Z depending on your settings.
Select your desired object from the object browser or click on it. If you forget this step, the following steps will crash Meshmixer
Edit, Plane Cut
Use translate tool to match “zero thickness” cut plane to “zero thickness” reference plane and apply.