Fight him like you would Dark Emperor Nyandam. He can now kill Dragon Cats in one hit and has more range, so he might take longer to beat. He has a really long attack animation (four entire seconds) and 3 knockbacks, allowing Awakened Bahamut to deal a lot of damage to him if timed properly. Yukimura is even more effective, especially with his True Form and Talents unlocked.
Tin Cat is an easily obtainable Rare Cat that can constantly reset his attack. This works especially well if you can push Kurosawah into the enemy base, allowing your Tin Cats to keep him pinned down. However, this will only work if Kurosawah has few to no other enemies protecting him.
Bombercat can easily stunlock Kurosawah, making dispatching him trivial. This is particularly effective on stages where Kurosawah is supported by other Black enemies, which Bombercat can also neutralize.
An effective strategy in some stages is to spam Eraser Cats and other meatshield to keep him back and then summoning Jamiera Cat to move in and put out damage. This works for a while until his magnification becomes greater than 300%. After that, your Jamiera Cat needs to be higher level.
Ring Girl Cat can heavily damage Director Kurosawah if she is high-leveled and Kurosawah isn't overly buffed. Her True Form does the job MUCH better though.
Guardian Gamereon can tank and eventually kill him, although an unprotected Kurosawa is a rare sight.
A small stack of anti-black units can constantly knockback Kurosawah with pure damage and prevent him from attacking.