This is where being in a HA property is actually a downside compared to a private rented property.
My DH has a property that he lets out and if there's any repair work to be done, they'll try and arrange it so that the tenants are there, but if they aren't in, the agents can arrange for it to be an unattended visit. If the tenant doesn't want it to be unattended, the agents arrange a visit around their timeframe.
I totally understand why the HA want someone to be there (ultimately they probably did do unattended visits in the past, but there is probably a history of accusations of theft, etc), but the inflexibility must be infuriating.
Do you have a housing officer you can speak/write to?
I'd respond their threat of "further action" with an email or letter saying:
"As you are aware, I live alone with my DC, one of whom has special educational needs [make sure you add this bit in as it will hopefully trigger something]. Furthermore I have no local family or friends that are able to be in the house while I take my DC to school. I have, therefore, no option but to take and collect my children from school myself, even on the days where your engineers are booked to call.
I have repeatedly told you this, yet the engineers continue to come when I'm not in and you won't allow me to leave a key somewhere for them to get in and start work before I return.
I have considered every possible option for being able to be at home within the time frames booked, but it is impossible for the reasons mentioned above.
Please do not continue to send engineers when I have told you I am not at home."