I think you have to check on your individual state or city laws. One place I know of requires that the LL send a certified letter to the tenant within 14 days of the tenant leaving. The letter has to itemize the deductions. If the letter isn't done or is after 14 days, the LL is not entitled to keep any of the deposit regardless of claimed damages. Another State that I've dealt in has something similar, but gives the LL 30 days.
This was handled in a strange manner. I can see where initially they thought the rug just needed cleaning, but after trying to clean they found it was ruined.
I think they messed up. I mean you have the money and have the upper hand in a way, but if you need them for a future reference this will hurt that possibility. And, it's possible that it could end up on your credit report as money owed.
Try talking with them and see what you can work out. In the very least, if you damaged the carpet and they are charging you, you shouldn't also be charged for the cleaning if it's a carpet in the same room.

I tend to think I'd ignore the letter but maybe you should first talk to them before spending more money on a lawyer. Might try to find out the laws in your state. It seems really strange that they gave you a partial deposit and now want more money. And I think it's their problem if they jumped the gun.

This is another classic example of the landlord screwing you for your deposit. If you don't pay the bill, they can screw you over by reporting it on your credit report and you'll have trouble getting another place in the future.