Focusing by moving a simple lens or an entire compound lens as a single unit will change the field of view. When the effect is seen in a camera lens, it's known as focus breathing. Some camera lenses, use "internal focusing", moving only certain groups within the compound lens, to avoid focus breathing. Since you're using a loupe and don't have control of the individual elements, the apparent change in focal length is likely unavoidable.

• I have not attempted focus stacking. However, I would consider using a mirrorless system with any internal-focus lens and extension tubes with pass-through electrical contacts.

• You can also check your focus stacking software for settings to align images despite being taken at slightly different focal lengths. For instance, align_image_stack has options to optimize for field of view, radial distortion, center shift, and XYZ position. Optimizing for field of view (-m) is the main option that appears to be applicable for focus stacking. Optimizing for Z position (-z) also seems as if it would be helpful. I would just try different options with reduced resolution images until I found something that seems to work.