Let's say someone goes through an invalid conversion, and now they incorrectly think they are Jewish. They pick a Jewish name for themselves, and they are called up to the Torah by that name and use it for whenever else the situation arises that it would be relevant to them.
If they get sick, and you wanted to daven for them or say a Mi Sheberach for them, could you / should you use the name they "picked" for themselves?
My question is not because of the Jewishness of the name. My question is because it is a) not their given name, or the name they go by regularly and b) taken under false pretenses.