Since the property is/was not listed with your MLS, why do you think you should put it in after the sale? If your agents are worried about "stats", shouldn't they keep their own.
Our Association of Realtors would allow my previous broker to submit production numbers to them from the office records (which kept tract of all sales, including FSBOs) at the end of each year and use that data for stats/awards, etc.
If the property wasn't listed in the MLS to begin with, I definitely wouldn't attempt to put it in after the fact. That would appear to give an inaccurate perception of MLS sales/inventory,etc.
...just my 2cents worth.