Bruce and The Justice League
Bruce will definitely surprise the League since they are used to "Brucie Wayne", who they may have seen in interviews (Clark), at galas (Oliver, Diana), and in the news. The fact that Bruce raised Nightwing will especially surprise them, thinking that there's not much depth to him, although they do know that he loves his kids and can be trusted with a big secret after Nightwing's reveal and his reaction.
There will be a bit of an exception with Oliver, who is scared of Bruce from seeing parts of the real Bruce when they were younger before "Brucie" was invented. He thinks that Bruce has just calmed down, but there are still moments where he feels intimidated and second guesses himself.
Things will change when the main League members are stuck with Bruce as civilians, trapped somewhere. It can probably be as a favor to Nightwing once they find out that Bruce is being targeted by a villain for something at Wayne Enterprises. I'm thinking the ones in the tv series mixed with the comics: Superman, Wonder Woman, Flash (Barry Allen or Wally West depending on the story), J'ohn, Green Arrow, and Green Lantern (John Stewart, although Hal would work too for comedic purposes). (I'll say that Hawkgirl already left at this point with the alien invasion, where Nightwing stood in Batman's place and didn't reveal his identity, but proved that he knew who they were. He's also not a full time league member since he's doing so much.) Bruce can get separated and trapped somewhere with Diana and Clark. Diana’s identity can be known at this point, leading to her taking charge of them getting out.
While they're together, they start to see signs of who Bruce really is. Bruce can flirt with Diana like he normally would, but then they slip into some more authentic banter like in the show. Clark can also get some sarcasm and drier wit from Bruce. They can also be second guessing themselves a bit because Bruce shows moments of him working hard and analyzing things, maybe figuring out who the bad guy trying to use his company is while simultaneously keeping his cover as an airhead socialite. Things will come to a head and Bruce’s mask will completely come off when they run into the bad guy; the villain captures them and incapacitate Diana’s powers, with Clark needing to hide his powers. They then threaten Tim or Dick, and Bruce goes off, escaping from the rope and starts fighting some of the henchmen, taking them off guard. He manages to get Diana’s powers back, which lets her take out the bad guy, and helps Clark get free, thoroughly showing everyone the man that could have been Batman.
By the end of the interaction, they are both impressed by and a little scared of Bruce, especially if he goes right back to his act in public, seeing how he raised Nightwing to be who he is.
Bonus for Bruce thanking them for looking after his eldest, revealing that he knows Clark is Superman, and also giving valuable information to the League that he stole from Lex Luthor in a meeting.