Hi Sophie, I identify as transgender, I discovered that late in life, and my wife is a cisfemale. We've gotten over the hump. Nearly all of our problems have one root cause: communication. I'm going to attempt to reframe your partner's behavior, and look at how you can address it in a constructive manner. My principal observation is that your relationship problems as you describe them have absolutely nothing to do with gender.
First up, the rugby game. Your partner was socialized as a male, and picked up some bad habits along the way. However, focusing on gender stereotypes is not productive! Their behavior is reprehensible, but I've heard this sort of complaint from cisfemale lesbians too. The real issue is about sharing the work load. Bringing up his gender will make him feel invalidated and judged, and you can expect the conversation to end in tears.
You need to have careful and respectful conversations about this, and your partner needs to meet you halfway. There should be some give and take, and it will go better if that is made explicit and nonjudgmental. Can you make a deal with your partner; he got to go out and have fun today, so now you get a turn to dump the kids on him?
Next up, jealousy and resentment.
One of the hardest things in relationships is emotional feedback. For example, it sounds like your partner is hyperfocused on his dysphoria (which is totally legit) and ciswomen are a big trigger for him. I'm guessing that you're one of the only people that knows? If that's the case, then you're the only person that he can talk about his feelings... and so you're the person that he complains to. The breakdown occurs because you're also the person that he's complaining about.
An example from my life: as I started dressing more feminine, my wife (a tomboy who enjoys being feminine sometimes) complained that she was worried about "measuring up" to my notion of femininity. What I heard was a complaint about me dressing feminine. She was expressing anxiety (totally legit) but when she described the trigger for that anxiety, she landed on one of my triggers.
My counselor introduced me to an interesting notion surrounding conversation: consent. It has taken some time, but my wife and I have reached a point where either of us can withdraw consent at any time. If she lands on a trigger, I can just say "I know that you're talking about your feelings, but you're describing me in a way that is hurting me. Can we not have this conversation?" She respects that kind of request, and she's started to employ the same tactic to protect herself. Sometimes it hurts to be shut down in that way, but it's a good thing, because we each have alternative outlets: our individual counselors, friends, family, etc., so we are still able express our emotions.
Of course, we don't want to mix this issue with the first one. If you are upset because your partner ditched you with the kids and you want to negotiate a fair trade, then they should hear you out. It's important to respect boundaries and it should be possible to withdraw consent from any conversation... but if he's unwilling to negotiate a fair compromise, he should expect there to be consequences for burying his head in the sand.
So, let's say that he flies off the handle because you're a ciswoman. In all likelihood, he's feeling very dysphoric on that day. He's upset, he wants what you have (which you didn't ask for, and which have biological consequences), and he complains about it. You interpret that as a complaint about you, or jealousy, or resentment. The truth is a mix of all of those things, but in all likelihood, your perception of that mix is to be more about you -- this is totally normal, we're programmed to think about ourselves first. I'm going to guess that you have responded to that in kind: you talk about your feelings caused by his words, and he hears a complaint, and perceives that complaint to be principally about his gender. Spiral to tears.
A way to address this is to say that the way that he's talking about his emotions is hurting you, and that you don't want to hear it. It can be useful to set a timer -- "I'm having trouble hearing this, can we come back to it in 20 minutes?" That's a softer approach, and it will give you both some time to cool down.
Ultimately, you need to be able to put your foot down. If he can't respect the withdrawal of consent, then you need to remove yourself from the situation -- in the worst case, that means packing up the kids and spending the night with a friend. He can stop that from happening in one way: agreeing not to talk about what he wants to talk about -- he'll probably spend the night sulking, but you won't have to hear it. If he physically tries to stop you from leaving, or gets emotionally nasty about it, then that's abusive and you need to treat it as such.
That said, it will go better if the first withdrawal of consent is not a surprise. Find a day when you're not upset, and your partner is not upset. Talk about consent and conversations in a generic way. Say that you've noticed this feedback cycle, and that you'd like to use the notion of consent as a tool to break the cycle. Try to describe an example of how him sharing his emotions (emphasize the legitimacy of his emotions) hurts you, and admit to being an imperfect human: you can't just hear that stuff without reading into it. Say that you need to be able to stop conversations in their tracks, and that when you've withdrawn consent, that needs to be respected. Offer him the same opportunity. Expect this to be a difficult conversation... but it will be loads easier than having it when you suddenly need it.
Edit: pronouns.
