Personally I would find a way to contact the president of the university (possibly through university PR, who also care about public image) and simply state,
"The registrar is asking me for quid pro quo, that I develop software for them in exchange to restore my ability to register for classes."
and include a screenshot of that communication.
Additionally, consider "agreeing" to their demands, if they will unblock you immediately. Register for classes, then reneg on your half of the "deal". Even if they then retaliate, that strengthens your position (a) that they are engaging in quid pro quo, and (b) that there's no valid reason that you should be barred from registering for courses, and also buys you some time.