I'm guessing the lights don't all turn on/off at the same time, there's really two or three strings woven together.
They do all turn on at the same time, but it definitely seems like at least two strings woven together
Three AAA batteries is only enough to light up two LEDs. The board must have a voltage booster circuit on it to raise the battery voltage to whatever's needed for the LED strings (something like a "joule thief").
Yeah, I think that's definitely it — I'm just at the frustrating point where I have a general idea of what to do, but not enough knowledge to know how to do it! I'll get there eventually, Thanks for your help!