If you can get the taxi to drop off at your CM even at none childminder days, and just pick your ds up from there, that might be your best work around.
Not all SN schools are suitable and many LAs work on a bums on seats attitude eg if they'll take them, job done.
My ds ended up in a 'temporary' ARP unit placement. He has ASD but many of the children there had a range of very violent and explosive behaviours which he found distressing (not to mention the language). He was there for years because the offers from the LA weren't suitable.
The day was shorter 9-3 so about 30 minutes shorter than the mainstream school and half days on Friday. So anyone who has a job that worked around school hours might struggle.
It is very hard to find out of school care or childminders who will take children with SNs.
While it is always the parents responsibility to find childcare, it harms no-one to acknowledge that it is a much tougher ask if a child has SN.
Unsurprisingly I've had to give up work and now home school my ds as there wasn't any other option.
While I don't think LAs set out to force parents of SN children to shoulder so much that mainstream parents rarely have to deal with, I don't think they are too bothered either.
I'm saving my LA between £25-35k a year by 'opting out'