AFAIK nuclear reactors are popular on naval warships because they dramatically reduce (and in some cases eliminate) the need to refuel. It's one huge factor crossed off of your logistics problems - modern CVNs can go literally decades without refueling, with the only resupply being jet fuel, munitions, and life support.
Doesn't fusion demand constant refueling (like most power plants)?
> A 1 GW fusion plant will need about 100 Kg of deuterium and 3 tons of natural lithium to operate for a whole year, generating about 7 billion kWh.
http://fusionforenergy.europa.eu/faq/
If it's possible to proportionally scale down that plant for the sub, then it would produce 50 MW for two decades on the same fuel. A nuclear sub weighs in the neighborhood of 10,000 tons, so 4 tons for fuel seems pretty easy to manage.
Obviously, all these numbers are very rough, but it's probably within an order of magnitude.
I don't think the half-life of deuterium/tritium is particularly long, so even if you could store it, it might not last decades before use. Though on the other hand, it's conceivable that you could design the reactor so that it produces deuterium/tritium on site from hydrogen cracked from seawater.
But realistically the reason we want fusion over fission in the first place is the proliferation concern. All of the other arguments against fission are NIMBY nonsense that could be applied equally to a whole swathe of chemical processing facilities that nobody seems to care anything about because they're not OMG nukular. And there is hardly a major proliferation risk from a reactor surrounded by military personnel on a vessel designed to securely carry actual nuclear bombs, so it makes the specifically military application kind of questionable.
We really need fusion for two reasons. 1) There is a better chance that we can actually build a large number of reactors without roving gangs of idiots destroying their cost effectiveness with gratuitous and intentional red tape, and 2) we can build them in countries we don't trust without making it easier for them to blow up the world, which will help to fight global warming vs. those countries continuing to burn oil and coal.
The difference between fission and fusion is that with fission you just load all of it into the reactor at once. With fusion you'll need a "fuel-tank" and feed the reactor continuously, but it's just a few atoms each time.
This is how I understood the concept, it may very well be wrong.
However, this is a distinction without a difference. Fusion fuel has an enormous energy density. For example, the fusion fuel capsule in a half megaton nuclear bomb is the size of a soda can, give or take. Deuterium-Tritium fuel has an energy density of around 3.4e14 Joules/kg or 94 million kilowatt-hours/kg. Even at a paltry 1% efficiency that means that operating a 50 megawatt output fusion reactor continuously for 10 years will only use about 50 kilograms of fuel, which is trivial to store.