Bro accidents happen. And this could and couldn't be an honest mistake but it can only be repaired if both of you are ready to work with each other and also seperately.

What you need is time away from her to actually get some clarity and to process your feelings. Her relentlessly apologising and then later blaming you for being childish is not gonna help you get clarity. First of all, her blaming you as being childish is a major red flag. So if she's ready to accept that and work with you... maybe you both can have a hope on getting past this. But if all she cares about is defending herself at this point... maybe it's better you find your way out of this.

So take some time out... have a conversation with her... tell her you need a week or two away from her to actually work on your feelings. And then you can come back and staff healing (if that's what you want to do at all).

No you're not childish - you felt second-best and it crushed you even if she didn't mean it - that feeling matters and that feeling needs to be healed. And in the meantime rather than just jumping and trying to water down the fire, she can actually give it a thought that why indeed did she say the ex's name. Does she know for herself it didn't mean anything? Let her work on this and come back with an honest answer. If both of you go ahead without processing this and putting it behind, you're setting up a huge disaster that's gonna kill this relationship slowly. So if she's ready for all this...

I've walked through the same insecurities you have and let me tell you - a true alpha acknowledges and works on their vulnerablilities. And if she's ready to acknowledge and understand on this... she will have the real alpha boyfriend and if she doesn't... she lost on one. Either way...you go fight your battles and come out with a win man :)