Sorry but no, the child can survive for a few hours without that pair of trainers (unless is it his one and only pair of shoes) if mum or dad are busy, out, working or in a date.
OP only asked her to pick them up “later” rather than straight away. Much like when they were married, he, she or the children should NOT be expected to drop everything and run to get whatever the other(s) need straight away, without, delay unless it is an emergency.
Divorce is not a marriage with two houses, the whole family needs to adapt to the new set up and its requirements and provide respect and privacy to each of their members.
Her having a huff and getting new shoes is highly unreasonable, I bet she would have made the child to wait if she was in the same situation.
I think OP she just got a bit of a shock to realise you were dating already and in the moment of shock reacted as she would have reacted if she was still your wife and you had told her you couldn’t deal with a request from her because you were having a coffee with a female colleague. How did you dare to tell your ex wife, the mother of your children, to wait rather than ending the date and run to do as wifey told you???? Again, this is part of getting used to the new set up and I’m sure in time she would realise this explosion was not about the children needs but about realising the level of influence she had over you is diminishing:
I remember my exh throwing a tantrum because I refused to order and pick up the food for a party he and his new partner were organising for HIS own birthday, he even shouted at me and told me I should do it because he was busy and new partner couldn’t help, even when we had been separated for years and both had our own new partners 😁
I suggest OP, for the sake of your children, let them have keys of the house so they can pick up stuff as needed if they are teenagers and you can trust them to knock first.
Don’t sweat the small stuff, just make sure you ask your kids if they have everything they need when they go back to mum’s house.