Atmega328 powered by a 3.7 volt battery?

I just know I'm going to make a persistence of vision display (POV). will I need 16Mhz for this?

I doubt it. A 328 running at 8 MHz should do fine. In which case, you will be able to run directly from the battery.

Do I even need an oscillator?

No. But the answer to the next question has to be "yes"...

Do you have a way to change the fuses?