It looks like the "Export License" option does what you want. I didn't know that existed!
Just export the license and copy it to your new computer.
As far as I understand it, there is no "license number" that you can just type into BTT, you need the license file.
As for the sync feature, look in the BetterTouchTool menu, click on "Settings", then look for "Sync" on the left side.
There are various other threads about this, for instance, take a look at this: