Are you absolutely sure that this is all about a twelve year old sitting and waiting for his mum to come home, in place he is regularly, with people he knows well and is perfectly safe with?
Because it really sounds more like the end of a shit rehearsal. One where all you want to do is go home, you can visualise exactly how it will be - and then it's different.
If you burn your boats over this, how easy is it to get alternative childcare for a 12 year old, at night, at weekends, for the duration of the run? They don't let kids scuttle around backstage anymore.
You will need to apologise for snapping - whether you mean it or not - to make sure you still have your childcare. But the chances are that she's used to your last week stress. So if you apologise, she's likely to accept it and you can get on with the show.
If you don't, however, well - doesn't look great to have 'yeah, was good, getting on a bit, heard she pulled out due to childcare issues just before technical on the last show'.