POV using ATmega328 powered with cr2032 cell
Not exactly an answer, but too big to place into the "comments" area.

Clock
For the 8MHz vs 16MHz, you should be able to run your project at 1MHz. Just use a bare AVR MCU with the Internal 8MHz oscillator divided by 8 - search for AVR Fuses. You can program it using an ISP (or and Arduino as ISP)

Battery
About the CR2032 or coin-cells in general are not designed to constantly supply current. You can use it for very low currents, up to 2-4mA or in pulses. Otherwise you damage the cell, reducing it's total usable capacity. Maybe have a look on sub AAA batteries instead of coin-cells.

Motor
For the motor, have a look on the micro coreless dc motors. They are really tiny and should offer an excellent efficiency. Motor example

Motor Driver
Not sure about the LM317. You don't want any kind of regulation as it would waste energy. To drive the motor a small Mosfet should do the trick. You can control the Mosfet using PWM, which will define the motor speed. Don't forget to implement some kind of rotary encoder to count the motor RPM so you can precise speed control - a tiny magnet and a hall-sensor should be enough.