As always I'm more than open to the idea that I'm not playing the match at a high enough level, but here are my 2 cents for what its worth:
-In a straight up pure zoning war I agree that the one with the life lead can maintain it if they play the game correctly. However, I also think its a lot easier for Batman to navigate through arrow's defensive walls than the other way around. Batman's Forward dash is really good both in terms of distance and recovery, and his trait can be a great tool to cover his way in. Add onto that your fire arrow are only doing .8% chip when he blocks them and you only get three before you have to take a risk reloading them and the fire arrow gauntlet isn't really that scary.
Now arrow navigating the batarang gauntlet on the other hand....its certainly doable but its harder IMO. Batarangs hit mid which means you either jump them or block them, and they do a lot more than .8% chip (especially if he meterburns them). Arrows jump is super floaty which can give him problems here....and trait is just as good defensively if not better than it is offensively. Then there's the final tool which leads to my second point:
-Batman's backdash is a problem for arrow IMO. Possibly his most underrated tool I think. Batman's wakeup game might be terrible, but doing oki on him can be a lot harder than it seems when he can backdash out of a ton of your options. Raw arrow beats it thankfully, but that gets beat by jumping. In general Batman's backdash is also really effective at getting out of arrow's pressure and frametraps also.
Overall I don't think the fight is too awful, especially not compared to some of arrows other matchups, but I would agree with it being 6-4.
It was probably my sibling 
Man I wonder what goes on in his brain in that situation 
