Solar Is Cheapest Electricity in History, U.S. DOE Aims to Cut Costs 60% by 2030

If solar were free, but we still needed to pay for battery storage, how would it then compare in cost to fuel-based alternatives (fossil fuel, nuclear etc)?

Would need $20/KWh battery storage to be competitive with nuclear for baseload according to https://www.cell.com/joule/fulltext/S2542-4351(19)30300-9 At the moment, we're at about $800/KWh.

Aren't car battery packs under $100/KWh? Is there some other factor which drives up the price for grid level storage?

The factor that drives up price for grid level storage is scale. Only ~300 GWh worth of batteries is produced globally each year. The world uses 2.5 TWh of electricity each hour. If anyone tries to install battery storage at a significant scale, demand will vastly outstrip supply and drive prices up.

Assuming there aren't economies of scale. Demand for solar had gone way up in recent decades (e.g. in germany, before it was cheap), and the price subsequently went down.

It's not clear batteries will do the same. While there's been effort to make batteries less reliant on scarce natural resources and mining, there's no guarantee we really get there. If we don't, price can be expected to go up, not down, with scale.

I would bet on price going down slightly with scale, but one can't really tell now what will happen: it might go up a lot, it might go down a lot, or it might stay flat.

There are thousands of different chemistries for batteries. The nuclear stans are betting that all of them fail.

None of those other battery chemistries are seeing the massive growth that lithium ion batteries have experienced. The nuclear "stans" are just pointing out that these are potential solutions, not actual solutions. If iron oxide batteries, or some other chemistry, suddenly becomes cheap and easily deployed at the TWh scale, great. But until then they're not a solution.