Reinstalling didn't help for me. And sorry for not answering earlier, I didn't get notifications.
The license files in question are located in C:\ProgramData\Electronic Arts\EA Services\License but I'd assume they're user-specific, so there wouldn't be any use if I'd just copy my file.
Other than that, the issues you're both describing are the very same I've had, constant popups, game itself starting when renaming the file.
As an alternative, temporary workaround, one could probably replace BF1Trial.exe with a dummy executable that does nothing, but the launcher would still run it over and over again.
I think the launcher simply shouldn't even try to activate the license, if the user isn't trying to actually launch the game (or trial in this case).
