The marginal cost of batteries grows more than linearly.
When batteries are covering 0% of the need, the marginally added battery cycles many times, so the cost is spread over many kwh produced.
When you add batteries to go from 99.98 to 99.99, the batteries cycle only for that 0.01, so the same cost to build them is spread over a much fewer kwh, making each kwh produced a lot more expensive.
Seasonal storage is madness: you charge once and discharge once per year. Pay 100$/kwh to install it, discharge 20 times (20 years, a 5% payback time, which is a bad investment), and you're paying those kwh a 5$/kwh premium on top of the cost of buying the discharged power. If the battery is instead installed to shift the production from 12.00 to 18.00, it cycles 365 times a year, so in 20 years the premium is 0.01$/kwh.
So nucleare doesn't compete with the first 40% of penetration of renewables and the first 30% of battery, it competes with the last 10%, which is still needed to get to 0.
Per Ember Energy reports, a cost optimal new build grid is between 90% - 97% solar/wind/battery, and between 3% to 10% gas peaker depending on how much sun/wind your locale gets.
But you can't replace the gas peaker with 3% or 10% nuclear because in essence that gas peaker is supplying 100% of the power 3% to 10% of the time.
So you'd have to build nuclear plants that can supply 100% of the power. But once you have that you might as well use nuclear power 100% of the time because the rest is irrelevant. But that's about as far from cost-optimal as you can get.
As you mentioned, using batteries for seasonal storage is madness. So to get to 100% carbon free you have at least 2 other options:
1. use a different form of seasonal storage. China is experimenting with this. In 2026 they will double the world's pumped hydro storage capacity. But doubling isn't a lot -- the world doesn't have much pumped hydro. But it does mean they might start doing it at China scale in a few years.
2. Overbuild to avoid the need for seasonal storage. Solar works on cloudy days and in the winter. It just doesn't work very well. So you need a lot of it. Which is expensive, but still a heck of a lot cheaper than batteries for seasonal storage.
In reality, most places will probably say that 95% or 99% carbon reduction is good enough and keep their backup natural gas generators around for the occasional dankelflaute.