Sounds like your sister is struggling and lashing out at a vulnerable time, by the time-honoured family method of being really unpleasant to someone she loves and trusts who is doing their best.
She could stand to take a breath and consider that she’s lived with this information a lot longer than you have, and she has had a lot more time to come to terms with that information.
The transphobic accusation is a red herring, which I wouldn’t dignify with a denial. You are understandably worried about extended family fall-out from this, and as an adult who lives in the world, your sister should know that reactions may not be uniformly positive and encouraging. The reasons for that can be as varied as the people involved and may be little or nothing to do with transphobia- case in point, you. It’s not fear or loathing of her, or of transpeople, it’s fear and loathing of conflict. Family reactions aren’t controllable by either of you, but will have consequences for both of you. Your DH’s family won’t be near the top of her list of things to care about, however, it’s very relevant to you.
I’d be inclined to say to her that while I support her right to exercise her choice, want her to be fulfilled and happy and love her very much, I am also a person with my own life and concerns and cannot be commanded to react according to someone else’s wishes or timescale. Presumably your sister knows first-hand how it feels to be put under pressure by the expectations of others.
That said, other adults’ reactions aren’t your responsibility either. You don’t need to ‘go against your parents to fight for this.’ Her transition is a fact, regardless of what they think, and they don’t have the right to control your interactions, nor influence your relationship.
Like other PPs, I’d say there’s no real percentage in keeping it a secret. You are entitled to love and support your sister as you see fit, and no-one gets to say otherwise. You don’t all have to agree. If ILs or DH or your parents can’t accept that you’re okay with it, that’s really their problem to solve, not yours.
As to your kids, they will likely be less perturbed than you think, and a brief age-appropriate explanation will hopefully be fine. I don’t know if you’re in the UK but I have a little kid who has known ‘sometimes people change from a man to a woman or a woman to a man’ for so long I can’t remember when we first discussed it. Your sister’s story is comparatively common now, and the concept has been societally normalised for kids. The likelihood is they will have trans peers either now or in the future, and the less fuss made of it the better for them as well - I’d point that out to anyone who’s openly hostile about your sister’s choices in front of the kids as well.