Note, your Melf's Acid Arrow example is dealing 5 points of damage on a miss on average. And it's available as a ranged attack.
Your Battlemaster is averaging 3 points of damage and cannot use a ranged attack. Nor, is he doubling his damage if he hits. Actually, the wizard's doing 2.5 times more damage on a hit.
If we doing the level/2 route, it would take 5 misses at 3rd level to equal 1 2nd level wizard spell. Seems pretty fair to me. By 6th level, I'd still need to miss twice just to equal one miss from a 2nd level wizard spell. Again, I'm not seeing the power problem here.
How many times is a fighter missing over the course of an adventuring day? Figure a 20 round adventuring day and about a 66% hit rate, he's getting 46 attacks, and missing about 15 times (give or take, presuming 3 actions surges and 6th level). So, he'd get a 45 point bump of damage over the course of an entire adventuring day. This is hardly going to break the game and it does bring the fighter up on par with the other classes.