All subs including big nuclear ones have battery. I think your question about battery is if the battery is used for propulsion. The answer is yes. The idea is to use the small reactor to constantly charge the battery. The propeller is driven by electrical motor from electrical grid that is fed by both battery and generator (driven by reactor).
The reason of using battery is because the reactor is low powered. The battery is acting as a cache for high speed. The reactor is to constantly charge the battery at low speed. The reason for low power reactor is its quietness and simplicity.
Good points. Also note that electric motor should be slightly more efficient. I think it's something like 30% efficient in thermal to energy. Then if you put the latest electric motor, you get maybe 90 to 95% efficiency on that. So 28% efficiency overall.
Not bad
So iirc
A really power diesel engine generates 2 mw power and a striking plant generates 320 kw power. So for a nuclear plant, you just need it to be like 2mwt to generate enough power for 560 kw energy and that can probably sustain 5 knots and faster on shorter bursts. It's possible my calculations are wrong here and you need something like 4 mwt to do that, but you get the idea.
To put things in perspective, a modern nuclear sun needs 200 mwt reactor.
With a 2 to 4 mwt reactor, everything should be quite simple.
What do you think
@latenlazy does that seem about right?