Battlefield 1 got just added recently to EA Access's list of games so you get unlimited hours nowadays as long as you got EA Access. Prior you only got a timed trial as a promotion sort of thing. So yes, it should work after it has finished up and it isn't just a trial anymore !
When the download has completed, if it still says that your trial has ended and just refuses to work, it might be that the former trial installation is getting in the way of the current one. In such a case I would suggest that you try and uninstall the game and install it back again fresh.
Sorry to hear what happened to your last trial @Melkins21. I hope everything would work correctly now after the download finishes.
Good luck !