How do I create a dynamic field on a front-end form with the Sprout Forms plugin?
Sprout Forms needs to know how a field should render on the front-end. By default it supports several common fields, but you will need to add support for custom fields on your own.  It sounds like you've got your field created and working, so you'll just need to add Sprout Forms front-end field support.

Adding support on the front-end is very similar to how you added support for your field in the Craft control panel. If you want to see several examples of custom fields adding front-end field support for Sprout Forms, take a look at the Sprout Fields plugin.