Well, I can't speak for AL because I don't play it. But nothing says an area that is "bright light" overpowers/cancels out/etc. an area of dim light. When I have the lights on, and light a candle, the light from the candle is still there.
Shadow Blade says, "In addition, when you use the sword to attack a target that is in dim light or darkness, you make the attack roll with advantage."
If it had said, "In addition, when you use the sword to attack a target that is in dim light or darkness, but not in bright light, you make the attack roll with advantage." I would agree with you.
It is like the weirdness of the invocation *Devil's Sight". It turns darkness into bright light, but doesn't affect dim light at all!
Such strange things happen in a world of magic and fantasy.![]()
All I can say is that I wouldn't allow it in my games. Basic logic indicates that you can't counter bright light with dim light unless there's magic involved. The book can't and doesn't try to spell out every contingency.
But if you and your group enjoy playing that way, more power to you!