Unlike Creative Suites, Lightroom serial numbers are usually multiplatform.
There should be no need for platform swapping.
Curious that the OP is saying that's not working on a Mac.
Actually, there have been quite a few reports of this problem recently. I can't provide links to any of the others because I don't know how to answer them and don't try. But there seems to be something wrong with the serial number verification system at Adobe.