I'm asthmatic, since a toddler myself, and so unfortunately is my poor little niece.
The hospital will not send home a child whose respiratory distress is extremely serious (though these days I admit services aren't what they should be) they test for blood oxygen, will give antibiotics, spacer, etc.
This is one of those situations where it comes down to how people cope.
It's being seen as an either/or scenario and as the partners desire to go on holiday as not working together and supporting his partner.
But if I were in the situation. I wouldn't necessarily see it that way. It can still be working together to say, toddler is ill, but not life threatening. You've planned this holiday, you go and when you come back, you hold the fort for a while and I get a break.
It's not wrong for the OP to request support and a termination of the holiday if she feels she cannot cope alone.
But it's also not necessarily wrong to say, 'I think I can manage this situation, go on holiday' if she feels she can cope. I think I would cope.
I don't think it's wrong that he still wants to go on holiday. I think it is wrong if cancelling the holiday is off the table altogether as a possibility if the OP requests it.
Family are increasing tension instead of encouraging unity. Encouraging resentment and a verses attitude, when it's mediating a relationship.