Jason,
You could feed the refnum into a Type Cast node (Advanced >> Data Manipulation) and cast it into a U32. I doubt that this number is guaranteed to be different with each connection, but maybe it will be in practice.
I would suggest an alternative approach of updating a state variable of some kind with a time stamp each time the connection is established/reestablished. Write that value to your logfile. Then, if you do run into a recurring problem, you can easily do some investigation to see whether a dropped connection is happening in conjunction with some scheduled event on one of your machines (e.g. virus scan kicking in at 4 AM, etc).
Regards,
John