I thought I'd give an update. I was forced to buy a new key/fob/remote from the dealer ($195). They wanted another $90 to program it for the door locks, etc. Since I wasn't sure my CLS would work with the remote, I declined. I got the new key and followed the instructions on this forum for programming the remote entry. Worked perfectly. Hazzah!!

My guess is that if the CLS works with the key, it probably is functional with a functioning remote. I know my old remote was broken because the LED did not light up when pressing any button.