Do I have to have The Legend of Zelda: Breath of the Wild DLC to keep the master sword on full energy? Because I have to keep going back and reclaiming it over and over again.
As part of The Master Trials DLC, there is a quest called Ex Trial of the Sword. After completing this quest, the Master Sword will retain is powered up state.
The DLC Wiki page states:
EX Trial of the Sword allows Link to increase the Master Sword's power; after the Trial of the Sword, the Sword will permanently retain its powered-up state with 60 Attack and 188 Durability.
Thus it sounds like you do need this DLC and you also need to complete some quests.
When you draw the master sword, it permanently gives you the Master Sword, which has its own slot.
When it runs out of durability, the message "The Master Sword has run out of energy" displays, and you have to wait ten minutes for it to recharge so you can use it again.
It starts at 30 damage and in the presence of a guardian or boss (not a mini-boss), or when you're in Hyrule Castle, it starts glowing, its attack power rises to 60 and its durability increases.
When you get 'The Master Trials', the first of two DLCs for BotW, you unlock the quest EX Trial of the Sword, which allows you to permanently upgrade the Master Sword in 10-damage increments by completing the Master Trials.
You access these trials from the pedestal in the Lost Woods where you draw the sword. Note that you can't do these trials when the master sword is out of energy. The trials are split into three segments, which can be taken on one at a time. Each segment takes away all of Link's equipment except for Key Items. This includes the Master Sword. You have to defeat every enemy in each room to progress, and the challenges aren't easy. It's like Eventide Island all over again. Make sure you have remote bombs + before starting this challenge. Also, note that you can take in a 30-minute attack buff as well as golden health and stamina.
I can't think of any reason why you would lose the master sword, unless you used some kind of glitch to drop it.