I think you've had some great advice - I strongly agree with the exam centre access, activity support, among other ideas brought up.
One thing I wanted to add if you're still reading is finding ways to connected with other departments. I know for me, I've had great help when I had malicious social service reports across both areas, but I think there are some missing tricks.
A light example - if the council welcomes schools to submit artwork or similar, it would be a good to include home educated kids too. It's a great way to include home educated kids in their wider community and build that trust that is sorely lacking.
A bigger example: young carers. These services have been cut to the bone for years, where I am a child basically has to be managing essential medications and doing a lot of personal care to qualify for actual support & even then it can be patchy. The rest are shuffled off to charities who, in my experience as a disabled parent, do fuck all (telling a disabled parent to paint their rental kitchen 'to cheer things up' as a response to my child has been bullied was not helpful). I wish there had been a way when I was trying to get support for my kids as young carers that it would have flagged something for the home education person because what we needed was access to social and emotional support for my kids as low level young carers, not people bringing us tins, telling us how to do up the house, and dismissing the bullying as 'something kids have to go through'. It really put me off reaching out for other services when this ended up upsetting my kids. I literally had people on new years eve showing up at my door shoving stuff at me 'because it's important to the donors' - I had stuff, and I'd rejected that Christmas stuff 3 times since we don't celebrate that holiday.
You likely can't do this, but I also have a wish that home education be included in school admissions applications online - just let us be in the drop down box and save everyone some time. Two of my kids, being registered as home educated since they were little, were given secondary school places and it was such a pain to cancel the places. I had to email repeatedly, had truant officers at my door that I had to explain that too, it was annoying that that department couldn't accept tht as I didn't have another school place that we didn't want that one & that there seemed to be a disconnect going on. Took months each time when I could have in a couple minutes happily filled in the online form, put home ed and we could all be on the same page with far less time wasted for everyone.
I had a fairly good relationship with my LA. I haven't heard from them in I think about six years - I cancelled one appointment because I was in the midst of funeral planning, and haven't been able to get in touch since (keeping the website up to date so messages don't bounce would also be useful).