How to keep the regulator cool while battery powering the arduino and accessories?
1W is near the limit of what a small LDO can handle. It will get considerably hot.

I would use 4 batteries instead of 5, and a 3.3V Arduino clone instead of the UNO, and power the LEDs directly from the batteries. You could also use your original UNO, feeding it with 6V on a 5V pin through a simple diode, which would drop it to something like 5.4..5.3V.