Go. Try to have fun. Eat cake. Be nice to everyone.
If I were the girlfriend, I would be extremely upset if you chose not to come to my birthday party. I’m also not sure how I would feel about the kids inviting my ex. After all, it’s my day, and i should get to spend it with who I want to. I’m kinda selfish like that.
Maybe the daughter thinks her father will simmer down once he’s met you?
If you are invited, you go. When there, be like someone someone would want as a boyfriend. That means no pouting that the kids’ father was invited. Also, it means showing that you are entirely secure about your relationship: she picked you to be with, so you don’t care if and for how long she talks to any ex. Last, if he starts talking trash about you (unlikely) or tries some one-upmanship (possibly very likely) you take it, and smile as enigmatically as you know how. Don’t engage - and so provide a nice contrast, and reinforce why she is with you.
ETA: If you don’t go, you’ll likely make it a pretty shitty birthday for her. Remember, it’s about her. Caring for someone sometimes means sucking it up.
Go. Be the perfect gentleman. It’s not “fake nice” it’s civility. Be civil. Show the manners your Mama taught you; or if she didn’t, find a list online and emulate them.
If he acts up, your only concern is her and her children’s welfare. Be there with the sole priority of making sure her day is as wonderful as possible. Gaze at her adoringly. Keep her glass full. Tell her how beautiful she is. Raise her spirits so high no one else can reach her.
Let him leave there knowing how short he fell of what she has now with you.
Suck it up. The world doesn’t revolve around you. It’s her birthday and her party. You go to it and you act nice to make her happy.
Alright then. I guess I’ll just suck it up. I’ll kill him with kindness. And try to make her day special as possible
I pretty much figured this is how this thread would turn out. I just needed the confirmation I guess.
Thanks guys.
As long as you’re not interpreting “kill with kindness” as being over-the-top sarcastic nice.
From your POV this guy is basically nobody. Any hint that you’re annoyed he’s there elevates his role.
No, I intend be kind to the guy. No passive aggressive BS.
Ok, just chekin’ 
These are excellent pieces of advice. 
Definitely go.
I do think it’s weird and kind of tin-eared for the daughter to be inviting her father to the party, especially knowing what he’s been saying about you. My parents are on good terms and I still wouldn’t invite my dad to my mum’s birthday party… unless I was hoping they’d get back together.
A. This trashing has reached you third hand, how can any mature adult take that seriously enough to rise to?
B. His ex, nor his kids seems to take any of it to mean much. Why would you want to give it an importance it doesn’t currently have?
C. Talking trash is so easy small children do it. Adults talk trash often with no intent beyond running their mouths.
D. Ignoring his action would make you look like the bigger man. Backing out will make you look like a weaker man.
You could try being ‘actually’ nice. Choosing to ignore what you’ve, ‘been told, by someone who heard it from…’, and resist judging someone based upon third hand info, until you’ve formed your own opinion, upon actually meeting them. After all, isn’t that how you’d want to be treated?
You were invited, so go. Spend more than you planned on her gift, giving her a reason to squeal a bit and hug and kiss you. Go out of your way to tell him what a nice card he chose.
ETA: if situations demand it, make an excuse to run out to your car and key his.
And if I were the daughter, I’d hand the book right back after I highlighted the introduction where it says the purpose of etiquette is to make people feel comfortable and at ease. I don’t see that the daughter has broken any rules. She might not have thought this through. She may just be hoping for all the adults in her life to get along. But I fail to see where she’s being rude. Handing someone a book to make them feel they’ve been out of line is both passive aggressive and rude, two traits that make me evaluate how much I want this person in my life.
Shakes, I’m glad your going. If my boyfriend ditched my birthday party and left me facing The Worst Mistake I Ever Made alone at MY birthday, I’d be pretty angry. You know she’d hear taunts about why you didn’t come. She doesn’t need that crash on her birthday.
I would go and show what a gentleman I was, and make conversation with the guy.
Be mature, and adult.
Find common ground.
Maybe talk about how great the sex is between you and his ex, and how she says that he was *the worst *in the sack.
Seriously, good luck and don’t give that guy too much thought. Just be there for your gf.
I agree 100%. I’m also wondering if the father didn’t finagle an invitation through the kid so he could 1) size up the “competition,” and 2) spend time with his ex-wife, maybe hoping to win her back. Both are pathetic so, yes, definitely kill him with kindness.
As others have said, this is about your girlfriend, not you or her ex. Ask her if she wants you there (I’m sure the answer will be “yes”). Then go, be charming and make sure she has a good time.
Nobody likes to be fake nice but we all do it. Suck it up for a few hours. You don’t have to be best bros.
“…This will gratify some people and astonish the rest.”
Well, I didn’t think it was possible to improve upon Mark Twain, but how about:
Always do right. This will gratify your friends and annoy your enemies.
After all, who cares about the unwashed “rest”? It’s those pesky enemies who should be irritated. 
It’s your GF’s birthday party, not yours, her ex’s, or her daughter’s. And as mentioned upthread, they’ll be cake.
Be a mensch. Go. And leave the chip on your shoulder back home. Or at least, in your car.