I understand your feelings about this all - they are fair. I think most of us would be offended and pissed at our DHs if they protected toxic BMs at the cost of their wives' happiness. 

I would continue to refuse to pay BMs soccer bill. If you pay it once, you'll pay it every year. Sometimes kids need to learn life in hard ways and SD not playing soccer might be a hard lesson for her, but in no way is this any of your fault. This is all on BM and DH. 

Around here, extracurricular activities are not considered part of child support, so DH would have to pay half. He does NOT have to pay BM's share - if she can't come up with it, then SD can't play.

I don't know how you make this work, he sounds like a big man-baby.  But I know that I sure as hell would stop paying for 75% of expenses so he can have extra "for the kids" if he's not spending it on the kids.

Are you opposed to working towards separating finances? Because holy heck I'd have a huge problem with my DH throwing a toddler tantrum regarding (the lack of) money to support HIS children's extracurricular activities.

He has options, like you pointed out and verbally gave him. Apparently, that's not good enough because he views you as a money train. 

"He has basically threw a toddler-like tantrum where he said things like “well I guess SD13 won’t play soccer then.”

"Well, I guess she won't be. That's unfortunate".
-- you

Separating finances sounds like a good idea IF your DH continues to pay his share. My concern would be that he'd be "short" and unable to pay his share of the bills because he's paying for things when BM gets her knickers in a twist.

I went through something similar. But the gist of it was the same. DH would agree to pay BM’s crazy demands and extravagant gifts for SD while leaving our household finances short. The grand finale came a month after the birth of our DD. DH lied to me and secretly paid bond on her rental and tickets for an overseas holiday in her gap year. Then accused me of hating his daughter and always trying to control their relationship when I traced the funds and confronted him. It made me feel like I’d wasted years of my life propping him up so that he could sacrifice me on the altar of the holy first family. In the end I decided to give him and our relationship another chance because DD was too young for me to cope with separation at that time. She was a needy newborn and I was braindead for the first year. 

I’ve since learnt that my DH operates on fear. When he felt like he had nothing to fear from me he was happy to throw me under the bus to avoid upsetting BM. My relationship with DH hasn’t been the same since. Huge trust and intimacy issues that hopefully will be improved with time. But it will never be the way it was. Once I couldn’t ever contemplate life without him. Now I’ve come to accept that there may be a day when I’ll leave him and I’ve made my peace with that. If I catch him at it again then I’m done and out. I guess in your case the decision is first whether or not to give the relationship another chance. But nothing will change if you don’t follow through with the consequences - even if the consequences means leaving him.

SD must play for a travel league. Sounds to me like a fair resolution for SD to keep playing and for you to not go into the poor house over it would be for her to play in your city/county rec league. I know here you are only looking at $50-100 for those. 

As for your husbands out burst- I don't know if I could get past that for a while either. You are entitled to your anger for sure! How to move past it? Time... Actions from him that place you as his partner and your marriage as his first priority. I would tell him that even though he apologized with words- he cut you too deep when he screamed at you. That you need time and you need action from him. Not silly flowers or a card- real action in your marriage. 

He apologized. And? What actions follow to show you that he is sincerely sorry? Or is the apology supposed to make everything go back to "normal"? I'd say you need to insist on a new normal, at the very least, where you are not expected to be the household ATM to cover expenses that he and BM can't/won't/don't want to.

But the bigger problem is DH thinking you should pay for his kids life,  SD has two parent, you are not one of them.  You are right if DH wants to pay for soccer camp, if it's that important, he should get off his ass and find a part time job to do that.  Not just sit there and think everyone ( you) should pay for his kid. 
No wonder this kid feels initial,  So does DH 

So he thinks you are his (and BMs) sugar momma? 

Awesome. Sorry, I already know that feels like crap. 

I also wanted to say be prepared for the guilt trips and don't fall for them. Telling you that you don't care about SD13, how selfish to not share, what kind of partner are you if you don't help, etc. Listen to your gut feeling. If it feels crappy and wrong it's because it is. It's a dual tactic of guilt trips and misdirection. With one statement like "you don't care about SD" he's immediately got your back up to prove that you do. Because you're a decent person and you do care. So you shove down the feeling of being screwed over to prove to him he's wrong. 

One thing that worked well with my DH:

If he said something manipulative, instead of getting defensive, I'd look him right in the eye and calmly state, "You are being manipulative."  His jaw drops.

Calmly ask him to explain to you why it should be your responsibility to pay for his kid because he and BM don't WANT to.

We redid our wills when we got married and DH didn't like how it was divided up. The lawyer explained that it was a standard division for second-marriage couples with kids (not split evenly between the kids). In a private moment DH turned to me and said that it wouldn't work because what would BM think when she found out about it?

There was a split second for DH before the mushroom cloud of my anger erupted. EXCUSE ME?! So what you're saying is that when you die that *I* have to take on your burden of worrying about BM? I don't think so and if you don't agree then we are just going to get divorced because I'm not budging on this and divorced it will be split just as it's written and the lawyer said is usual. And how in the hell would she even find out how it was split? Are you gonna "tattle" on me to her and exactly what sort of control does she have over me? She may have control over you but there is absolutely no scenario in which she has control over me.