Problem of 2 copies of Starcraft

I bought my copy of Starcraft when it was first released 10 years ago. Recently I registered it (to get an extra chance to enter into the beta opt-in). Then I bought another digital copy of Starcraft, partly to get one more chance of opt-in, partly to allow friends of mine to come to my house and relax.

So I downloaded the new copy (or I thought so) and installed on one of my computers. Yesterday when I tried to logon to battle.net on both copies, I got the message that the CD key is been used. Turned out that I install the same copy on both computers.

I thought it will be a easy fix, just change the CD key right?

Well, I found it is impossible to find the CD key, either from the installed product nor the installation file. To solve that myself, means I need to uninstall both copies, re-download both copies (because I don't know which copy I downloaded), and re-install them (the downloaded digital copy don't ask for CD keys, it appears to be embedded in).

I have sent an support email to Blizzard. The best alternative I found on the internet is a CD key changer, but it is third party and I don't want to risk my CD keys.