Supplying power to an 5V LED strip
While it's not possible to turn all the LEDs on max brightness without extra power, the main problem was that I hadn't realized that Pololu sells 2 almost identical 0.5m LED strips, and I was reading the instructions for the other variant.

Now that I'm sending instructions using the correct protocol everything works fine as long as I limit the number of LEDs I turn on.

Sorry for the mistake.