Hi, Albat, and welcome to the FICO Forums!
Sorry to hear you're having trouble with your issuer. Unfortunately, what the rep told you is correct. Also, your CR will only show the CL where it currently stands, not what it used to be.
What the CCC is doing is known as "chasing balances." They are not-so-subtly attempting to push you to pay off your card as soon as you can. This type of behavior is common among CC issuers when they are uncomfortable with the amount of time a cardholder is taking to pay off a balance, and it is their way of attempting to push you out the door.
Unfortunately, once you make the final payment and the card is PIF, you should probably expect that they will close the account. Many members here, when they have found themselves in that situation, have paid the CC off as soon as they could and then immediately closed it themselves, so that at the very least their reports would reflect "closed by consumer" rather than "closed by creditor."