Powering a 3 LED strip with AAA?
The linked LEDs are designed for 12V not 5V. You would be better off with 5V smart leds. You can under drive the 5V ws2811 LEDs to about 3.7 volts.

Yes 3xAAA would be enough to power 3 of the 5V LEDs at a decent brightness. Preferably you want Alkaline not Heavy duty (aka zinc) batteries for a longer life. Alkaline AAA would have about 2000 mAh, so assuming full white 60 mA for 3 LEDs that's 180 mA so roughly 10 hours per battery set. That value is optimistic, but if you have different colors and brightness and off periods the run time will increase.

You could go smaller with lithium cells but make sure you get a protection circuit and a charging circuit. There are tiny lithium packs that are the size of match books. Those are typically 3.6v so you may need a step up voltage converter so that adds complexity and lowers the battery life. You would basically be making a usb power bank.

So your best bet is to go with a small usb power bank. I've had one that was the size of car key fob.