I won't comment on your relationship per se but I will comment on your partner's desire to send your daughter to private school as I think you need to talk to him and give him a stark reality check.
If you cannot afford a car, how on earth will you afford school fees? Has your partner actually looked at the level of fees? If you can scrape together the fees you then have to pay for the uniform, the school trips, the meals, so many extras and so it goes on. Each year gets more expensive too - over and above inflation.
Also if his income is unpredictable and you can't pay the fees one year do you intend to shift your daughter between the private and state sector in accordance with income flow?
Has he considered how your daughter will feel when her friends are dropped off in the 4x4 and go on holiday at least once, if not twice, a year? I know life shouldn't be about keeping-up-with-the-Joneses but I think it's hard if you're the one that's left out.
Wouldn't it be better to look into where there is a good state school and work towards moving into that catchment area? I honestly don't think that private primary school achieves that much when weighed up against the cost and at the local primary your daughter will be able to make local friends.
Maybe once she gets to secondary school age your finances will have stabilised so that private school can be considered then where it may (only may as it depends on what she wants to do) impact her choices IF she wants to go to university.
It just seems very pie-in-the-sky to me on his part based on the facts you've presented here. Perhaps he's saving the pennies as I type which is why you're so hard up now...