I don't think it sounds as if you can afford it properly OP.
Send your daughter to local outstanding infants and reconsider at the next stage. At this age it's nice to have very local friend who can just pop round for tea after school or go to the local park together.
FWIW I've always been advised (on here and elsewhere) that if you can't afford the full 14 years private, saving your money for years 10 & 11 is the way to go. At secondary level, at today's prices, that's a minimum of £25k. For just those two years.
I was at private school in the days when fees were not exorbitant (no fancy facilities!). One of my best friends had to leave at the end of year 9 as her parents couldn't afford it any longer. They had never had enough money but had somehow hoped it would be OK. My friend was devastated to move at this stage. Now we are adults she talks about how damaging it was to her relationship with her parents at the time.
You can't bank on your daughter getting into a grammar school or getting bursaries at secondary level.
If you are both working full time, how do you intend to cover childcare in the longer private school holidays? All the costs at private schools are higher - lunch, uniform, trips, extra items for extra curricula stuff.
Do you sums very carefully, on a spreadsheet, and factor in the huge increases in fees every year, and included everything.