Is the relationship of the questioner considered forbidden, and what should she do if her husband refuses to rectify the contract which she considers invalid?

If your marriage contract is invalid—as previously explained—then it is not permissible for you to remain with your husband before renewing the legally valid contract. If he refuses to renew it because he believes the original contract is valid, then you should both present the matter to trustworthy scholars so they can explain the Islamic ruling to him. If they rule that the marriage is invalid and he still refuses to renew the contract, then you must separate from him, either through divorce (ṭalāq) or khulʿ (a form of divorce initiated by the wife).