I'm going to conditionally agree with feejer222, at least for states in the U.S. In some jurisdictions, accepting the refund of the tire mounting price by cashing their check can be considered an "accord and satisfaction" of a disputed claim. In others, it has to be clear that was the intent, or in many (like California, by virtue of statute) requires a writing to the effect that the intent of the payment is to fully discharge liability for the disputed claim. (BTW, it must be a disputed claim to function as an accord and satisfaction. If there is no dispute that X dollars are owed, in other words, if it is a "liquidated claim", then a payment of less than X dollars is only an installment -- not what this situation sounds like.)
On the other hand, you might be able to get them to give you a check, or to offer that in writing with or without a check, make a photocopy of it, and then write them a letter stating that the check is uncashed (and it needs to be) and offering to return it. Your letter might state your reasons (e.g., what you think they owe you with calculations) and ask if they want you to cash the check as an installment on what you believe they owe you or return it. That way, you at least get the evidence, and maybe an installment payment.**
** Disclaimer: I'm not admitted by the bar to practice law in your jurisdiction, don't know all the facts, you should see a lawyer for competent advice on this issue in your jurisdiction, etc., AND this is not guaranteed to work, but it's what I'd do here. That is -- it's what I'd do if I didn't mount my own tires. This kind of F&^% up is a perfect example of the number one reason I do my own: I know it will be done right, and care will be taken to shield the rims from damage, even to the extent of using duct tape and cut out cardboard where advisable (like on the custom aftermarket rims on my XX).
