Maybe I am misunderstanding but it seems to me there are 2 separate issues here. Firstly how to cover the travel day - one day per month you're proposing - when you are inevitably going to need tonnes of wraparound. This I think you would need a regular contract with a professional for - and I think that is achievable. But it would need to be scheduled not ad hoc. I reckon once a fortnight is quite doable for this if not once a month, but talk to some agencies.
The issue of child not attending school can happen any day and I'm not sure exactly how that fits in. It sounds like you'd feel able to cover that 19 days out of 20 by WFH, is that right? Or are you looking for more childcare outwith the 1 day a month because you're already (understandably) at breaking point trying to juggle picking up your SEN child with your work, even when WFH?
You have a huge amount on your plate, but at risk of adding to it, it sounds like your SEN child's needs are not being met in school and that puts you rather on a knife edge. These situations get worse more often than they spontaneously resolve. It doesn't sound like you have a set up with much "head room" for your child to cope with an extra demand like you being away more. A child on a PT timetable can easily become a child out of school, and what would you do then? There are never any good answers to this because having a child not in school is an incredibly difficult position, especially for a single parent. And it ends careers.
If SEN child's school is not working anyway, maybe a house move wouldn't be a bad thing. It would give you the freedom to relocate with their schooling in mind. But I understand the need to stay close to a support network too. Would a fully WFH role actually solve the problem you have, or do you actually need a bit more headroom anyway, even if that is just paying a nanny one day a week, or a couple of afternoons? I think this would take the pressure off. The nanny could pick up the other child from school a couple of times a week too, get their dinner sorted etc which would give you a little breathing space. A friend of mine employs a housekeeper who is around when her secondary age kids get home from school, does a few hours' cleaning a week, laundry, preps a couple of family dinners.
With your current employer, do talk to work about whether they can make an adjustment bearing in mind your child's needs. A good employer should be very careful not to discriminate against you for having a disabled child.