If you are trying to detect whether the relay is open or closed, then yes, you could use the relay contacts between CTS and serial ground (double check this) to detect whether the contacts or open or closed.
I don't understand the serial number part. I don't know how you are trying to read its serial number based on a single CTS binary input.
The comand I'm sending is telling the device to read its serial number and it is a 16 digit binary number, then I have a nother comand that alowes me to select a position in the serial number string and depending if it is a 0 or 1 the relay then closes or stays open. So I then let my program go throw all 16 positions an reads the binary code with help of the CTS.