A fingerprint on a modern lens is usually not a disaster. Modern lens coatings are generally quite resistant to skin oils, so the risk of permanent damage is low, though not zero. On some older lenses, fingerprints have been known to etch or stain coatings over time, which is why it’s best not to leave oily residue on the glass for months if you can avoid it.

A lens pen used with light pressure is unlikely to damage the coating just from pressure alone. The bigger issue is that an old or contaminated lens pen may just smear oil around instead of removing it fully.

Best practice: clean it properly with lens cleaning fluid and lens tissue or a soft lint-free cloth. Put a small amount of fluid on the tissue or cloth, not directly on the lens, and wipe gently from the center outward. If you see streaks, use less fluid and finish with a dry clean part of the cloth or tissue.

So: likely no serious harm, but clean it properly when you can and avoid aggressive rubbing or using a dirty lens pen routinely.