I'm not sure how applicable my experience will be.
I sent all three of my kids to private high school. One of them went on to a private (out of state, FAFSA based) university. The other two went to in state public universities. One of the kids also went to private K, 3rd, and 4th.
I also, FWIW, attended public K-12, an Ivy, out of state private, and in state public university.
Generally private schools reevaluate the family's financial situation each year. FAFSA is based on prior prior year, but it's still a year-by-year evaluation. They do this because family financial situations can change, sometimes significantly, over time and that's the fairest way to do it for all of the families involved at that school.
It's probably going to end up being a sliding scale for you, with income being considered more heavily than assets since most schools figure on being paid out of cash flow. So the more you make, the more you'll pay. K-12 private schools are expensive and consider themselves a luxury item, so what they think you can afford will probably be more than what you think is affordable.
Most private schools have discounts for second, third, etc. kids in case that might apply.
Most K-12 schools increase in price as the kids get older, and sports and extracurricular activities probably come with additional fees. It would be wise to look at the price tag for all 12 years before you start.
Public K-12 schools have better gifted and talented programs than private, which is a bit of a surprise, but it's because there are more kids and they'll have enough to have a separate class. There can also be more options at public K-12 simply because there are more kids. One of the private HS my kids went to only had maybe 1/4 of the sports of public, no band, and very limited orchestra.
On the other hand, private K-12 the kids can get kicked out for bad behavior, so that is less. Also, most kids at private K-12 at least have parental pressure to do well and be college bound. In smaller K-12 private schools, there can be more individualized attention and more unique types of education - my middle one went to an IB school and has his IB diploma.
If you want financial assistance, you can expect to fill out that PFS or equivalent form every year. In my experience, it'll be due every year the summer before school starts and probably is based on the prior calendar year tax return and current balance sheet. If you're getting significant aid, it would not surprise me if you went through a verification process where they confirm everything (this happens to my kids with FAFSA frequently).
If you're looking at multiple kids, or private university and/or grad school for the one, I'd take a look at whether or not you think you can and will choose to afford private school the whole way through for everyone or not. If not, perhaps consider where private school makes the most difference. My personal opinion is that private K-6 or so is probably not a good value, especially for gifted kids. It's feasible to switch them into private for 7-12 and the high school years are probably the best bang for the buck because that's what colleges look at.
Overall in my personal experience and evaluating that of my sisters, my three kids, and my six nephews and nieces, private is better on an absolute basis but not on a bang-for-the-buck basis. In my case the Ivy was a particularly poor value due to it being a poor fit for me (lots of culture shock).
Private universities regularly give out $10K or $20K or so in "scholarships" which are really just tuition discounts / price differentiators / sliding scales. The private high schools I knew didn't do this, although one of them did seem to have a policy of about 5% of the student body being "charity cases" where the families couldn't afford to send their kids there but the kids were unique or special in some way and so the school effectively sponsored them.
Of course, everyone around you will have a variety of opinions on all of this. You have my 2 cents, worth what you paid for it probably.
