I'm confused. Part of it may be terminology.
What I'm about to say is on the "Purchase" link above, but just for clarity's sake:
When you purchased, you should have received a Registration Code. It is your permanent license. You can use it on our Restore Page up to ten times a year.
Each time you do a restore, you will receive an Install Key, which is good for 14 days.
We can't e-mail you these, and if you're having a problem now, it's important to find the cause since it will continue to cause problems in the future.
Either you aren't receiving e-mail from buy-button at jriver OR there is just some confusion.
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I believe you did receive the Install Key. That's what you're executing when you say "when I click to open it I merely get the J River screen". That's what should happen. There is nothing useful inside the file. Its filetype (mjr) is owned by JRiver Media Center, so MC will open it, find the license and install it automatically.
Save that file on a USB drive or similar and transfer it to your other PC. Execute it there by clicking on it from Explorer.