Can I apply for a tourist visa extension with less than 15 days left on my stay in Thailand?

you can apply for an extension on your tourist visa when you have 30 days or less left on your stamp at most immigration offices. <- that means you can do it right up until your current stamp expires if you want to

That 15 days left on the stamp is to APPLY for a Non-Immigrant visa inside the country and has nothing to do with extensions on visa exempt or tourist visa entries