How to keep the regulator cool while battery powering the arduino and accessories? I'm trying to build a battery operated hand lantern. It has the Arduino, an MPU6050 accelerometer, and 4 APA106 (WS2812B-like) LEDs.

If everything's powered through the Arduino's 5V pins, that's 80mA for the Arduino, 240mA for the 4 LEDs, and let's round up to 5 mA for the MPU6050.

If I'm using 5x AA batteries for 7.5V (at full charge) that should be 2.5V * 325mA of power wasted or just under a watt that'll be heating up the regulator? I'm not sure if that's huge or tiny or in between or what.
What's the most sane way to power this device for an absolute newbie?
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.