It sounds like the reason your creditor will not negotiate your interest rate is because your credit card is near, at or above the high credit limit, thus reducing your credit score, which makes you a higher risk. You could think about asking them to freeze your account, which would also freeze the interest charges as well. It will close the account, however, when you do this. If you are having a hardship but want to make the debt good, some creditors will work with you. If you do not get the answer you are looking for with the first call, then keep calling or ask for a supervisor/manager who can help you. Do not get mean and hateful with them on the phone, that will turn them off from helping you. If you can show you genuinely want to satisfy the debt, someone will work with you.
Good luck. I hope this helps.