Current all-in battery costs are about $200/kWh. Daily global energy use is about 400 TWh. That would be an 80 trillion dollar battery for just one day of backup.

That just isn't feasible to build near term and it isn't even enough to prevent blackouts. Plus the current methods probably don't scale that far, so the cost is even higher.

Previous generation of nuclear reactors in France were built for less than 2 billion euros (constant euro) per GW (sorry it's in french, you can see the numbers here http://i-tese.cea.fr/_files/LettreItese18/ECLAIRAGES/REP.pdf on page 10). There were 50 reactors of three types (900, 1300 then 1450 GW). If we had kept this technology instead of losing the know-how and developing a new program, we could certainly build 1500 GW for those 3T. Don't know if any other country ever achieved such economy scale on its civil nuclear program, maybe South-Korea as like France they have only one nuclear company, and even more reactors per sites of few different models.

Finally and (slightly) anti-nuclear piece worth reading. Yes, solar is easy for capitalism, but I'm not sure building all those batteries is. Building smaller pre-fab nukes vs building larger battery arrays seem pretty close in terms of challenges.

You don't have a citation on "$3T would buy 200 GW of nuclear power plants", and I very much doubt that extrapolation, because if any sane planner got the $3T budget, the first thing they would do is invest heavily in pre-fab.