Topic on Talk:Battlefield 4
Yes, borderless would be the ideal explanation once it's working (unless they implement second monitor support in-game for this in the the final version which isn't likely).
I don't have the beta to test but is it possible to put the browser window on the correct monitor ahead of time, or does that make the game then start on that monitor? If it can be moved before launching that would be the better way of wording the solution.