Your poor future SIL!
Getting engaged is clearly a big deal to her and your brother, they wanted to announce their news with all their close family, and that clearly doesn't include your boyfriend and his part time kids.
You said yourself you only met his daughters 6 months ago, and he's not their residential parent. They're not even a stable part of YOUR life, let alone your familys life.
It was perfectly reasonable therefore, that even if it HAD just been a casual family lunch, that they wouldn't be invited.
You were too busy being offended on his kids behalf, even though they almost certainly didnt care, to consider anyone but yourself.
You were told sorry, we cannot accomodate them. You pushed, and were told sorry but no, a second time.
You tried to go around her and undermine her by messaging your brother to badger him for an invite for your boyfriend and his kids, and were told again no, the kids cannot come, and he did relent to possibly your boyfriend attending but presumably you told him not possible without the kids too.
You then, doing the "it's all about me" dance, went to your mum complaining and crying, she tried to help by contacting your brother, and he called you, clearly annoyed, and told you again, NO.
To suggest he should have informed you this was a special occasion as way of placating you, would have completely defeated the point of a surprise announcement lunch and ruined it.
You made your choice not to go, they were probably disappointed by that but nothing they could do without spoiling it. You took it upon yourself to decide WHY they weren't invited, and that your desire for your extended family to meet your boyfriends kids, trumped their reason for not inviting them. You should have hosted your OWN event, not hijacked theirs as a meet and greet just because it was convenient.
You had no idea what time they would actually be eating, just the start time of the invite. You may have assumed correctly the meal was over, but that doesn't mean its suddenly ok for you to show up, as it was the EVENT they werent invited to, not just the eating part.
You rang the bell, nobody answered, so you seriously just took it upon yourself to trespass on to the property and go round to the back garden?!
They had literally JUST announced to the garden full of close family that they were engaged, and at this moment, in a case of worst timing ever, you, your boyfriend they've barely met, and 2 kids they have never met, come traipsing round the corner, announcing your presence/arrival.
All that planning, their anticipation, their excitement, and the reaction to what to them is major news, some of the biggest of their lives even, was completely spoiled by everyones attention being suddenly directed to you bowling in with YOUR agenda.
I totally understand her running off crying, this was her big moment. If she doesnt have children especially, then this is the biggest thing that has ever happened to her. All she wanted was the people she loved most there to celebrate that, and she didn't get any of the happy reaction and congratulations like she had no doubt been imagining/dreaming of because you came crashing right in the middle of it.
They had been undeniably clear with you NOT to bring your boyfriend and his kids, and you did exactly that regardless. You've taken the attention off her at one of the key memorable moments of her life, shes devastated. I don't blame her for banning you from her wedding, how selfish and self centred you've acted about this, ignoring everyone elses feelings and wishes and doing what YOU wanted at the expense of everyone else, shes probably convinced you'll steal the limelight at the wedding too.
Give her time and space, don't push yourself on her, don't bleat on with apologies. Write a letter, then keep your distance and let her come to you in her own time if she chooses to. Don't go in for any big public shows of remorse, you'll just be making it even more about you.
Your boyfriend deserves some blame too, if he knew him and his daughters hadn't been invited. He never should have gone along with just turning up, especially when the door wasn't answered where you could have been politely turned away.
His older daughter sounds nice, but the younger one doing impressions of your future SIL is incredibly mean, thats bullying and is disgusting, making fun of someone in distress.