In the USB tab of Configuration Utility, check the box which says "Use serial number string" and then you can enter the required serial number to that specific part.
Please have a look at the attached snippet of configuration utility.
The CY_DEVICE_INFO structure and CyGetDeviceInfo or CyGetDeviceInfoVidPid API can be used to retrieve the serial number of a part in an application by passing the device number or vidPid of the device of interest. Please refer to USB-Serial API Guide section 3.4, 4.4 and 4.5.
We tried again, as you suggested, but this time we cycled the power after writing the serial number and before checking to see if the serial number was there. The serial number was programmed as we wanted. Certainly, our experience with using Cypress's USB-Serial Config Utility to configure a new chip from its factory default to I2C operation is that the PID does not change as shown in Config Utility until after the power is cycled (a lesson learned the hard way). Based on this recent experience with the serial number, it appears that the power must be cycled before the config utility shows the newly programmed serial number. Whatever the details are, it's working for us now. Thank you for confirming the process. I will mark this question as "Answered".