Is there a way to populate a drop-down with the templates files in a folder?
This would be a good task for a plugin.

You'd essentially need to build the plugin to do a couple of things:

(1) Use PHP in a service to list the contents of a directory. Once you have that list, you'd then need to...

(2) Deliver that information to the front end via a custom field type. You can then use your new custom field type wherever you'd like.