Submarines do have a large lead-acid battery for emergency power. In fact, the WWII-era boats had bigger and better batteries.

However the battery isn't there to start the Emergency Diesel Generator. Submarine EDGs are air started, for many very good reasons.

We still need the batteries since it's possible to suffer a loss of all AC power while the boat is submerged far beyond feasible snorkel depth, and have to recover from it without broaching.