Though both WMI classes are scanned and generally return the same serial, I do believe the serial visible in the Summary tab of Windows computer webpages is actually pulled from win32_computersystemproduct, not win32_bios. The serial pulled from win32_bios is listed in the Config\Hardware\Bios section of the computer's Lansweeper webpage instead. If neither tag is the one you're interested in, the only way to detect additional tags would be through
custom registry scanning. You can always manually add data to your assets as well, as explained by Daniel.
Changing the warranty scanning procedure, which is hard coded into the Lansweeper service, isn't possible. Warranty scanning takes the model, serial and (sometimes) SKU visible in the Summary tab of the asset and submits these values on the vendor's website to retrieve warranty details.