I think you build a network for the non vomitting times they are ill and need to be off. Other parents may be willing to have a child with heavy cold or recovering at end of chicken pox, if not sickness and diarrhoea. This then reduces the time needed off to a more acceptable level. Offer holiday cover in return.
Why was the child off each of the previous times? Could someone else have had them for one or more of those slots if you had something in place?
I think schools are a bit more understanding too if you are very apologetic and make clear in all communication that you are doing your best to arrange alternatives and sharing cover with your DH too. If you show you have made all efforts for this occasion and as a general principle, so are taking your obligation to TRY to put in place alternative arrangements, it will be more difficult to take it further. Being defensive, suggesting only you can carefor the child, indicating that partner has not done any cover, that no efforts have been made to find alternatives even if this is hard, are all likely to go against you.
So I would get on the books of an emergency ychildminder or nanny service if there are really no adults around that you could arrange to call on in an emergency. And I would work hard to get one or two mums lined up, even if they can only be available 1 day a week. And I'd make an arrangement with relatives who may be hundreds of miles away, that if an illness looks like going beyond 2 days (is chicken pox) that they would come. When relatives know this will be rare, they often will agree and this withhold your DH can reduce absence to just 1 day from 3 or 4.
Doing this stuff wil ease your mind and if you can tell your HR manager you are doing it or have done it, will help. They will have some action poi ts to write in their fi!e to indicate the issue is being addressed.
These don't help with the initial absence but can for future, as can saying the rich stuff in the meeting. Its the pits isn't it, but this phase won't last forever and you just need to get through it.
Additionally, of course never keep child off unless absolutely necessary. Vital to save any time for the the illnesses which really need you to be off.
Realise this was before Christmas.
What happened in that meeting? Hope all are well now!