Where a nuclear reactor comes ahead is the fuel cost. Which is zero (it's included in the construction cost, and there's no refueling for the lifetime of the ship). A conventional turbine burns fuel, and over the lifetime of a ship it can add up to quite a lot. It can certainly get above $1BN.
In fact, the CBO analysis shows that the total cost for 5 such ships would be $14.2 BN using conventional engines, and $14.8 BN using nukes ([4] page 6 top table). That was when the oil price was $86/barrel. Now it's $110, so just because of that the calculus can switch to favor the nukes.
[1] https://en.wikipedia.org/wiki/America-class_amphibious_assau...
[2] https://en.wikipedia.org/wiki/General_Electric_LM2500
[3] http://www.fi-powerweb.com/Engine/Industrial/GE-LM2500.html
[4] https://www.cbo.gov/sites/default/files/112th-congress-2011-...
Later the Navy decided that they did indeed want nuclear powered aircraft carriers, and so now they have them.
And how do we know that for sure, given the army tendency to cover up wrongdoing?
https://en.wikipedia.org/wiki/United_States_war_crimes
https://en.wikipedia.org/wiki/List_of_nuclear_power_accident...
https://en.wikipedia.org/wiki/Ocean_disposal_of_radioactive_...
Not to mention the societal risk in having critical civic infrastructure controlled by the military.
Two US nuclear submarines have gone down, the Thresher and the Scorpion. Neither was due to the nuclear power plant, and so far they have kept the radiation contained.
That says everything.
If the containment on these reactors was lost, the effect would be measurable due to the sensitivity of such detectors, but literally a drop in the bucket of waste disposed of into the ocean.
This 2011 CBO analysis [1] looked into the feasibility of outfitting US Navy surface ships with nuclear reactors. The US Navy is known for burning more fuel than some small nations. The conclusion of the study was that it's not worth it, but they did not consider CO2 emissions at all.
On page 10 this study implies that the cost of building an A1B nuclear reactor (one of the 2 powering a new Gerald Ford carrier) is about $0.9BN. This cost is classified, so this number is as close as you can get to the real cost. The power generation capacity is also classified, but you can infer it could be about 400 MW. So, that's a cost of about $2.5 BN/ GW.
This cost is not available to everybody, but it certainly is available to the US Navy. It's conceivable the lawyers could find some loopholes to make it available to the Department of Energy as well.
So, if Congress is serious about climate change, they can easily mandate the US Navy and the DoE to order a bunch of A1B power plants, and use them to replace conventional power plants or provide clean electricity. There's simply very little that prevents Congress to do that. The technology is there, there's nothing to innovate. The economics make sense. The safety of the US naval reactors is simply unparalleled. In 70 years of service they did not have a single incident. The manufacturing capacity is there; the US is already making about one such reactor per year, they can increase that to 10 per year in a decade without any issues.
[1] https://www.cbo.gov/sites/default/files/112th-congress-2011-...