If you electively home educate, you take on responsibility for education. Whether you educate directly or outsource, and whether you educate at home or outside the home. DD won’t be on the roll of a school, there won’t be a school providing work.
With flexischooling, you would need them school to agree. You take on responsibility for the education outside of the timescales DD spends at school. Again you can do this directly or outsource and it can be at home or not. The school doesn’t have to provide work.
Rather than the above, if DD is unable to attend school full-time, under section 19 of the Education Act 1996, the LA has a duty to ensure she still receives a suitable full-time education. Then the LA remains responsible for this education. Contrary to what some LAs say, consultant level evidence is not required and schools don’t have to send work home and even if they do, it doesn’t relieve the LA’s duty. The form this education takes varies. It may be at home or not. It may be online or not. It may be 1:1 or not. But if you EHE, you relieve the LA of their duty to provide this.
Does DD have an EHCP? If not, you can request an EHCNA yourself using the model letter on IPSEA’s website. If she already has an EHCP, have you requested an early review? With an EHCP, there are other options.