[Nabeshin]

I am not an expert and rather new so if someone more experienced can advise on this specific question please do so.

But I would heavily advise against providing a different serial number than the actual one you received. (I don't have any direct experience for this exact scenario but I will provide reasoning why).
Most companies when making a sale will record the serial number of the device. For example, when Amazon ships Apple products, e.g Airpods, the serial number for the airpods is included in the order information. Similarly, I don't doubt that the retailer you are trying to SE keeps some record of the SN of the item sent to you. It is not safe to assume the opposite and do what you said.
E.g they could keep some sort of a record like this, Order #2412451, Item Name/s: Apple Watch Item/s SN: 25012051050
So if you provide a different SN that has the potential to complicate things a lot in my opinion.

This may also be the case, but then why bother asking for the photo of the SN?