Fortunately, it appears to be possible to retrieve the serial number if you have access to your Nike+ FuelBand account.
To do that, open the Account Settings from the top-right drop-down menu and, after the resulting page loads, click on the Nike+ Devices tab on the left. After that page loads, view the HTML page source and search for serialNo. It should take you to what appears to be a JavaScript object notation (JSON) structure that reveals your FuelBand's serial number to the right of the serialNo text. There may be several serialNo strings, and the one that appears to be correct is the one that is numeric and not those that are alphanumeric. The serial number doesn't appear to get projected into the user interface, but it's at least within the web page source.
This is likely to be the correct serial number, and the settlement form allowed me to use that as the serial number. That serial number was different for the two FuelBands that I bought during the permitted timeframe, and it seems likely that it's the serial number that they are looking for. It's worth trying out, at least..