Ah Iceland! I see that your support options may be limited.
You at least can try to submit an online support request (See the link in my post above).
Regarding your problem, the only thing I can think of is the Logic purchase got hidden somehow. Yes this is possible!
First: On the new Mac, in the Mac App Store check if the Logic purchase appears in the Purchase History.
If it is there then try to unhide it:
Hide App Store purchases - Apple Support
Hope that works... Beyond that, I'm at lost. Really trying to help.
WD