OP, I think you have to think carefully about transitions both into and out of the arrangement that you are requesting, and about exactly how you want it organised.
(FWIW, we had a somewhat similar situation with DS, who actively suffered in his first primary due to being an outlier in ability. For us, the solution was temporary home schooling followed by a school - and house - move. Don't underestimate the difference between apparently similar schools - both DS's primary schools were Ofsted rated Good state primaries. One was so bad for him he ended up with physical symptoms from anxiety, the other he absolutely flew in)
IME both as a parent and a teacher, school weeks have a particular 'rhythm' - sometimes one of timetabling, sometimes because the teacher plans in week-long blocks. This is more apparent in higher years, but is still there in lower years. So for example, always missing a Friday might mean that you always miss the 'final piece of writing' on a particular topic, or tend to miss problem solving in maths, or when most 1;1 reading is done, or maybe miss science or games, or the celebration assembly in which awards are given out and achievements recognised.
So even though your child may be 'way ahead', always missing a particular day may mean that they miss a particular part of the planned learning - and the school's communal life - on a consistent basis.
Also think about when you might want to go back to full time schooling - is this an arrangement that you want for e.g. the remainder of this academic year? For the remainder of KS1? For the whole of primary school?
I think what might be most likely to be approved - and might avoid some of the main disadvantages - might be one or two afternoons a week, rather than one or more whole days. If this was to attend a particular 'other educational experience', the school may also be able to use 'non absence' register codes (e.g. educated off-site, or educational visit / trip) which may make the inevitable impact of 80% attendance on the school's absence statistics more palatable.
Do think carefully about the school's point of view - you want your DD to have a positive experience when she is in school, so understanding their viewpoint and working towards a compromise that works well for you all will be important. If you can't find a positive way forward, then consider the 'whole hog' alternatives - home schooling; moving schools - while also thinking about how e.g. playdates, activities such as dance or Rainbows can maintain her existing social links.