In North Carolina, a liquidated damages provision [i.e. you get no refund clause in their contract] is binding and enforceable where it is not intended to act as a penalty. See City of Kinston v. Suddreth, 266 N.C. 618, 620, 146 S.E.2d 660, 662 (1966).
“A stipulated sum is for liquidated damages only (1) where the damages which the parties reasonably anticipate are difficult to ascertain because of their indefiniteness or uncertainty and (2) where the amount stipulated is either a reasonable estimate of the damages which would probably be caused by a breach or is reasonably proportionate to the damages which have actually been caused by the breach.” E. Carolina Internal Med. v. Faidas, 149 N.C. App. 940, 945−46, 564 S.E.2d 53, 56 (2002) (emphasis in original).
To determine whether a provision is a liquidated damages clause or a penalty, the Court looks to the “nature of the contract, and its words, and tr[ies] to ascertain the intentions of the parties . . . .” Knutton v. Cofield, 273 N.C. 355, 361, 160 S.E.2d 29, 34 (1968).
In your case, what you would argue is that their clause saying you are entitled to NO money back is unenforceable because the clause acts like a penalty rather than a liquidated damages provision. For example, if their profit on the $6,000 procedure is, say $1,000, then they should refund the other 5k since no costs were incurred because you opted out of the procedure.
Using these arguments should get you a substantial refund. If they do not work with you, you may need to file a declaratory judgment action against them and ask for a court order to get a ruling that the clause is a unenforceable penalty.