How do you think Bruce would react if he found out he is going to fight the Joker forever? That their relationship is going to last longer than any other villain? I'd imagine it to be equal parts horrifying but also gratifying, since on the one hand, the Joker won't stop trying to hurt him, on the other hand, he has all the time in the world to figure out a way to fix him or at least come to some form of an understanding that can pacify the Joker or maybe even make him an ally.
Well... The thing is, he doesn't have to find out. Bruce, as canonically depicted in multiple recent stories, has always known. From the very first meeting.
But way more explicit than this, there's the rewrite of Bruce and Joker's first ever meeting by Tom King in The Winning Card, told in Batman: Brave and the Bold. It's been referenced as current canon by the New History of the DCU #2, and directly by Zdarsky's Batman run in Joker: Zero Year. Basically, Zero Year happens right before The Winning Card.
Now, is The Winning Card good? Much like Tom King's writing, it's... uh... it's something. There's a lot of elements I like and also a whole bunch of things I hate. But what I did enjoy a lot was how (in the spirit of The Man Who Laughed which was previously canon) Bruce was a complete freak about Joker and pretty much kickstarted Joker's obsession with him. Joker kicks Bruce's ass and, stubborn bastard who hates to lose that he is, Bruce fixates on beating Joker's ass back. And in their final confrontation...
"I'm willing to keep going... if you are." [looks into the camera]
And then Bruce proceeds to tell the world's worst joke (ugh, bat-tery), and it's so bad it's got Joker judging him. He nearly stabs Joker with the knife Joker stabbed him with previously, because King has a kink for stabbing as penetration (I am not kidding, check out Batman/Catwoman 2021 #9). Afterwards, Alfred has this insight to share:
The story explicitly frames Joker's incessant telling of jokes as a way to try to connect. "Do you get it?", he keeps asking. Joker's dialogue is always inside these beautiful vintage frames, not speech bubbles; he's not potrayed as human. He's not a person, he's death. But Bruce is the only one who even recognizes what Joker is doing as an attempt to connect, and is the only who tries to understand. He tries to speak to Joker in his own language, and at the end, Joker responds like a normal person (in speech bubbles) only to Bruce.
The knowledge that him and Joker are forever comes up in other stories too, by Morrison and Snyder specifically. So Bruce has always known, on some level. Combined with The War of Jokes and Riddles that takes place in the second year of Bruce's career, also by King, Bruce not only knows, but also believes that he owes who he is to Joker, in some capacity (because Joker stopped him from stabbing the Riddler; see what I mean about King having a thing for Bruce stabbing people and it being a sex metaphor? Joker really didn't want Bruce penetrating killing anyone else). And yet, it takes Bruce almost a decade to try and avert the lethal ending of their relationship in The Killing Joke. Years before that, he refuses to let Joker die after he got the death sentence in Devil's Advocate, when it would've been so easy to allow it. All along, ever since the beginning, Bruce has been constantly saving Joker's life-- not trying to permanently disable him or put him out of commission without killing him, not trying to fix him or rehabilitate him ever again. Just to give some examples... for someone who got over his fear of guns to shoot Darkseid (Final Crisis #6) or who's dealt with an undead villain by condemning him to a "fate worse than death", sending him off into space to float there forever (Batman Incorporated 2011 #2), it doesn't feel like he's ever tried that hard, does it?