In my case, I bought my original game from Steam a couple of years ago (which is why it runs directly from a launcher rather than via Steam, this has since been changed), plus additional accounts directly from Frontier.
I didn't then need to "reinstall" the game, merely copying my existing folder was enough. I then activated a launcher in each new copy using the email address I'd used for each Frontier purchase.
If you've run multiple CMDR's on the same PC before, but not synchronously, you will probably have shortcuts that now point to the wrong place: to additional launchers in your original installation folder, rather than to the new folder. The location of the launcher in your file structure matters.
I didn't then need to "reinstall" the game, merely copying my existing folder was enough. I then activated a launcher in each new copy using the email address I'd used for each Frontier purchase.
If you've run multiple CMDR's on the same PC before, but not synchronously, you will probably have shortcuts that now point to the wrong place: to additional launchers in your original installation folder, rather than to the new folder. The location of the launcher in your file structure matters.