I'm not a mum yet myself, but I do have BPD and something I saw in a Facebook video that seemed genuine the other day, lined up almost exactly with how I'm taught to handle BPD emotional flare ups- I can't find the video to link it atm - I'll post back if I do, but in the meantime I'll summarise as best I can (It probably won't be word for word lol). Perhaps this may help you?
A mum was making a cold coffee and getting her toddler involved to help - super wholesome video. After it was done, her child pushed/slid the freshly made coffee forwards on the counter, it spilled of course, and made a massive mess.
The mum was clearly pissed off and angry, but instead of saying or doing anything, she just stayed quiet.
It was super difficult, and you could see it was insanely hard for her as it was a really big mess and she visibly needed that morning coffee, but she powered through with keeping quiet and just trying to collect her thoughts and calm herself down before she responded.
In the time she was quiet, her toddler responded by; initially being giggly/silly because funny big mess and unsure how to respond, then going quiet too and mimicking mum, then saying "oh no", then realising on their own that they had an accident, and apologising.
At this point, the mum was still angry, but had calmed down enough to talk at a level volume, and say "It's okay sweetie, but see when you push things that hard, they'll fall over, and then we have a big mess" or something alone those lines.
The toddler then said "Did you want to help me clean it?"
And the mum said "Yeah, of course I'll help you clean it up"
They then cleaned, mum still coming down off the emotions I think blurted out "Mama really needed that coffee this morning" under a sigh.
The toddler then apologised again, and she said "It's okay, accident's happen"
This is the exact same thing I'm taught to try and do when I'm having an especially tough time emotionally to stop me from lashing out.
Just take a second, breathe, collect your thoughts. Take as much time as you need, remove yourself from the situation if you have to if noise is overstimulating. Think about why you're SO emotionally reactive in this moment, has it just been a rough day and things have piled up, or is it justified? Has something genuinely enraging happened? Try and ground yourself outside of your anger.
If it's the first one, think about what went wrong in your day and why, and realise that it's likely not anyone's fault, life is just poo sometimes, and that you just need to take a step back before responding.
Think through how you'll respond in a way that's calm and kind, even if your child overtly did something wrong, they're a child, they're allowed, they're learning.
And most importantly, if you slip up? Take accountability for it.
Everyone is fallible.
No matter who it is, and especially if it's your child, own up to your shit.
Explain you've had a rough day, and you're sorry you're taking it out on them and that you didn't mean to, and that you're trying really hard to do better because you know it's not okay.
It will also teach your child coping mechanisms over time if they see you doing this.
That the appropriate way to handle your emotions is not to have an outburst, or tantrum, or yell, or scream, or be aggressive. But instead to take time to calm down, think about how they're feeling, and come back to things with a level head. To talk, rather than to fight.
And beyond that, it also teaches them that if they DO slip up, that they should take accountability as well, own it, and apologise where necessary.
Talking things through is infinitely better than giving into the adrenaline of your emotions, even if you feel justified in the moment.
As someone with BPD, I unfortunately know this excruciatingly well.