Hi there.
For clarity, only licenses granted via Frontier store orders will show up on your account under 'My downloadable products'.
The transfer tool simply adds a license, without the need for an order behind that. So your transferred licenses will not appear here.
You can download the standalone launcher here:
https://hosting.zaonce.net/elite/Client-Installer.exe
If you would prefer to run the game via Steam instead, this FAQ outlines how to claim a partner key for your Steam library:
https://customersupport.frontier.co.uk/hc/en-us/articles/360011813740
If there are any further issues though, please just let us know!
-Luna