Completely avoided this thread until today, when I was finally able to see the film.
Initially, I was worried at the opening dancing scene. But my concern proved baseless. I loved this movie. I laughed, squeezed out a tear or two, may even have Grooted in my seat a time or two.
Didn't really consider the battery subject before now, because it was a part of the character that I really enjoyed, but seeing as Peter was on a pirate ship, it's possible he was able to find a comparable substitute when the batteries eventually died. Pirates loot, and when they can't loot, they salvage.
Also, if Rocket can figure out how to cobble together a (if you saw the movie you know what) from scrap in a short amount of time, it figures that Peter, having the motivation of someone stuck somewhere for a long time, could figure out a power source from available materials. Likewise for the tape deck in his craft.
*shrugs*
It worked for me, and that's what was important to my viewing experience.