The Mk-48 ADCAP torpedo does require prep time, they aren't generally kept in a "ready to fire" situation all the time.
I would imagine one simple solution would be a capacitor, since the battery has to be good in storage for long periods of time, but when actually needed, only has to work for between 10 minutes an hour. So the torpedo has some spin up time while the battery charges the capacitor, however it is also possible to get its initial charge from the launching vessel (while in the tube), they are wire-guided after all so in the tube, they are "plugged in" to the ship.
Why an hour of battery when the run time of the motor is 7 minutes? Because in a combat situation, the captain may well order torpedoes loaded and they may sit in the tubes ready to go, so the batteries are running. It is possible that if they drain the tubes and pull them out, the batteries have to be replaced.
These are all minor considerations, considering that each Mk-48 ADCAP costs $3.5 million dollars each, they can (or should) be able to afford both the best batteries as well as spares and replacements.
Also worth noting is that a torpedo is not a small weapon, the modern versions being over 19 feet long.