There's no need to buy the full version. LR is cross-platform.
The serial number for the stand-alone version LR6 is the same for Mac and Windows.
If you have a CC subscription, you can log in from either Mac or Windows.
All you have to do is to install LR on the platform you need it and make sure to sign/log out on the platform you do not need.
Download from here: https://helpx.adobe.com/lightroom/kb/lightroom-downloads.html