My guess is that the League and Court are truest of eqauls in combat skill but if the court has one advantage in combat as a whole - it’s their immortal soldiers that stay awake for 24 hours before having to get recharged again. May sound like a flaw but 24 hours is more than enough to carry out the plan set beforehand. Even a Lazarath pit in Gotham wouldn’t recharge a person as fast as this the Courts tech.
You make alot of fair points in your comment but I disagree with you on one thing… When you said Batman and Robin have more of a home field advantage, they don’t.
How do we know this? We know this because The Court Of Owls has far more knowledge of Gotham’s layout. An example would be a bunch of hide-outs, and other hidden where abouts that Batman could never find all his years, it proves that Batman isn’t as familiar with Gotham as he’d like and if he isn’t then the Bat-Family isn’t and if the bat-family isn’t, then the League isn’t.
This all comes from the Court’s deep ever-present roots within Gotham, their resources and history within its premise, they’ve been controlling Gotham for generations, they were aware of Batman’s location (the batcave) for years. The world’s greatest detective wasn’t able to find the Court Of Owls in a lifetime, they had to find Damien, then Bruce which is the only reason he was able to track them, if it wasn’t for that, he still never would have had a clue - despite fighting them at the owl hideout where he got defeated by the undead talons . So I wouldn’t say that Batman or the Bat-Family know most things going on in Gotham, not as much as we’d like to think anyway. With all the training and resources at Batman’s disposal, he would have never gotten close, not sure how much the League knows about the court since that hasn’t been expounded on nor explored but I know that the Court is well aware of the League since literally nothing goes on in Gotham without the Court knowing and pulling some strings in the background while they’re watching.
But yeah, besides all that, Gotham is fairly large, and since Batman is far from being the only hero who takes to the streets, being fully aware of Bruce Wayne (Batman) and his activities/actions isn’t to be 100% fully aware of Gotham. I highly doubt that the League knows about the court’s existence but have prepared for war and while the Court knows about the League, they wouldn’t care about them unless they became a problem in their city. That’s what makes this battle so interesting imo
I’m going to guess that if League had any real advantage, it would be in their versatile weaponry, keep in mind that stealth and smoke didn’t do so well against the Court and any type of gas is pretty much rendered useless against the Court’s undead assassins as far we can tell. I suggest explosive long range arrows would be way to go if the League is to get a foothold.