Bateman:
How can I take you seriously without a business card?
A reference to a wonderful scene in AP. Where first Bateman shows his business card and then everyone 1-ups him with a better one and he narrates about how mad it is making him. This is almost like the start of him going mad insane.
I stomp a bat like a dog, there's no turning back, I've gone too far!
He stomped on a dog's head multiple times to kill it 
You're vile, you reek of failure and you're stupid
Before he killed a homeless man he told him he was vile, reeked of many things including piss and shit, called him stupid, etc. Reference to that.
I serve up a Psycho filet, so don't just stare, eat it
He's a psycho/it's in the movie title, he is shown eating filet's a couple of times in the movie. One of his most famous quotes in the movie was when he had 2 women prostitutes and wanted them to have sex with eachother, in the middle of a rant about Nick Collins he sees one prostitute just staring at the other from behind and says "Sabrina don't just stare at it, eat it". I think you know what that means.
A mindless vigilante can't step to the true AP!
Batman is considered a mindless vigilante by a lot in the Batman series, AP stands for American Psycho
I break your hero spirit like Bane, and faster than Luis turned gay for me!
Bane had the whole quote in TDKR: "I wondered what would break first.. your spirit.. or your body *breaks back*". It's a cool quote. And it works here for sure, and Luis was a character in AP who Bateman attempted to kill but Luis mistook it for a homosexual encounter and said he had always wanted Bateman.
You can't handle my evil thoughts, just like you can't handle your parents death
Bruce Wayne still can not really accept the death of Thomas and Martha Wayne.
You always disappear out of thin air, who does that remind me of? Oh, Paul Allen.
Paul Allen was Bateman's first "big" kill, and he made everyone else think he just vanished out of thin air without telling anyone. The detective had also thought that. Batman often just vanishes into a new place or area.
Batman:
You had one too many Dark Nights, I'm here to bring justice
Pun on Dark Knight, most of Bateman's killing was at night.
My crave for stomping scum is bigger than your blood lust!
Bateman says several times in the movie he has an incredible crave for blood lust.
When it comes to battle rapping, you fail like your movie's sequel
American Psycho 2 with Mila Kunis was considered a huge fail, and most fans of AP don't even want to acknowledge it's existence. :P
You've gone mad Pat, like your mind my raps are lethal
Kind of a reference to the theory of it all being imagined as well.
Killing prostitutes and homeless men is really bottom of the barrel
Bateman mainly killed prostitutes and homeless men, bottom of the barrel is an expression.
Your life's in shambles and needs to be repaired, so let me call Harold
Harold Allnut is a person in the Batman comics who repairs Batman's equipment.
It doesn't matter what happens inside, you didn't really do it
There's a theory about the American Psycho ending, that he didn't actually do any of it and he imagined it all.
What makes you really insane is your love for Huey Lewis!
Bateman really loved Huey Lewis and while he is by some, he's not considered great by all.
Bateman:
You deal with Heath Ledger? Jack Nicholson? Ask my lawyer, I'm not Joking around
Both played the Joker, pun on it/joking
Bruce Wayne's got a date with the devil, NOT A FUCKING CLOWN!!
Bateman is a very evil person and serious, he's not a clown like the Joker
You've tried to 1-up and step on me for too long, us rich guys always win
[b]Reference to Bateman saying in the movie he's tired of getting 1-up'd, and in the end when nothing happens to him it's the fact that rich guys will win because they don't want their social afflictions to be effected because of one person (Bateman)[/s]
Although my favorite really has to be Going Back, by Phil Collins
Bateman CONSTANTLY talks about Phil Collins and his songs, even when having sex for example. No matter what. Going Back is a song by Phil Collins, and it's a double meaning saying he should go back/get off of his toes.
Batman:
You're a bland boring man from the 80s, what's new?
Bateman is just another suit from the 80s, other than what happens in his mind.
90% of the time the people you were with didn't even know it was you!
A lot of the time someone would be talking with Bateman and not even know it's him, confusing him for another suit because they all look alike. Sometimes even talking about Bateman "behind his back"
Well in my mind you're the kitty, I'm the ATM
Bateman imagined an ATM saying "feed me a stray cat" and tried to insert a kitty into one, it's basically just saying Batman will chew him up. AKA beat him AKA bully him AKA whatever.
There's the batsignal so I'm out, time to deal with another bag of shit
I thought it'd be funny to make Batman swear. And also Batman constantly has to deal with criminals, every few minutes maybe