Dell has two utilities available for download that may or may not help with installing the Service Tag number. I have listed them below and you can give them a try. The utilities are not for all Dell systems and in the event they do not work on your model you will have to contact Dell Tech Support again. After you explain the problem with the motherboard replacement and the need for installing the Service Tag number, Tech Support should send you out the right utility. Since the Service Tag number is basically the serial number of the machine, it should be installed in the event of possible re-sale and Dell should have no problem with that reasoning.
Click here for a download of SVCTAG.EXE which is a Service Tag utility.
Click here for a download of ASSET.COM which can be used to install the Service Tag using the following command switches.....
asset /s [XXXXXXX] Sets either a 5 or 7 digit code. X's being the tag. (Try using both a small and Capitol "S" if one switch doesn't work)