Will doing a chargeback on tuition hurt my chances at getting into the same university for grad school?
Would this hurt my chances at getting into the grad school later on?


Practices vary, but probably not.


  Will it hurt me later on?


Very likely.  You probably will not be able to complete a graduate degree if the university thinks you owe money.


  I should just let myself be robbed?


No, you should insist on a refund.  Ask the instructor and their department to help you get a refund.  Ask the registration department again.  Ask to speak to the head of that department.  

HR is probably the wrong department to contact.