Just transfer the installer across to the new computer. So many ways to do that ... network them, burn to disk, via external disk, portable disk, send via a host site, etc.

If you purchased Pro from the 'Official' site, then you will see your url personal download page link from (a) the email they sent you, (b) from your invoice. See below image in my first post here in the asterix: Upgrading to V 4.18 (click on link)

BTW I find best solution to LW installation problems on OSX is re-install the system. A fresh install, deleting previous system (not archive install!) so.. if you try that, back up your files first.