Hi Mark,
I see 3 ways how you could realize this (more or less):
1. Use the self enrolment plugin and in your course use as first and mandatory activity the feedback plugin in combination with the conditional activity feature. So you would make all subsequent activities dependent on the completion of that feedback activity.
2. Use my plugin 'enrolment upon approval' (moodle plugin database). There you can let users fill in a text box before they enroll in a course. But the admin/teacher would need to confirm course enrollments (that's the original intention of this plugin).
3. You create a new plugin which hast the exact features which you need. Feel free to contact me if you need help with that.
Flotter