After getting some help at an Arduino/Raspberry Pi user group Meetup, I've made some significant progress implementing the keyboard scanning. Now I can get it to output the correct button that I've pressed.

Wiring Issue Fixed

When I first plugged it in, I was getting garbage from my keyboard scanning algorithm (nested loops energizing a row and reading each column). It was registering a lot of nonsense periodically, basically noise. It turns out I neglected to include a pull-down resistor on the ends of the columns, which meant my values were floating, causing it to randomly trigger. After soldering on some 10k resistors the problem went away entirely.

Debugging Issue Fixed

I was also having trouble debugging with the Keyboard library because it would spit garbage out into my IDE. Someone at the Meetup pointed out that I could leave off the Keyboard library until I really needed it. I feel a bit dumb for not thinking of that, but it worked beautifully.

Next Steps

So now I need to implement some debouncing, likely just a time delay, and some way to save the state of the keyboard so I don't send a bunch of the same key press every time. That all feels pretty doable, so that's what I'll be doing in the next couple of weeks during my spare time.