I’m sorry about your mum.
In these posts, you’re cross with BIL and SIL equally it seems to me, but although they had met your mum, really to your SIL she was:
Her husband’s brother’s sister’s mother.
Yes, in Irish funeral terms that puts you high up. My parents report back on funerals they’ve attended in NI and I know what you mean about the culture! But not every single person in Ireland is the same.
That’s not a close relationship, and your BIL did acknowledge your loss, in responding to the initial communication. For most distant relationship (husband’s brother’s sister’s mother) I think it’s normal for one person’s response to be “for the family”.
I think it’s shit that BIL didn’t send a card too, but I don’t see why SIL should take the flak for that.
You said yourself that you didn’t attend her dad’s funeral. You had good reason with the young children, but why can’t she be allowed her own reason not to travel 100 miles to your mum’s funeral?
Even the fb outpourings over her own dad... just because you behave that way for your own family doesn’t mean you have to do things for other people’s families. If she had posted in fb about your mum, would that not have got your back up that she was going OTT?
I get that it’s disappointing for you. You don’t sound like you like them, even before this. So - don’t go to their event. I’d not be going simply because I didn’t like them, it wouldn’t be because the lack of contact after your mum’s death. They could have behaved a lot better - but not everyone does cards and supportive messages. If you don’t want friends who don’t - phase them and their event out. But put the blame on your BIL more than your SIL, given the distance of her relationship with your mum.