The first order of business is to figure out the connections on the joystick. Luckily, these work exactly as you'd expect. The black wire is common, and the other four are the direction switches. I used the meter to determine which was which when the joystick is sitting in my desired orientation, then labeled them for later.
Next, I attached the headers that come with the bluetooth module. I like to solder in a cross-pattern, the way you would tighten bolts on a sealing surface. This keeps the header from getting crooked, and spreads out the heat load as you go. Note that I'm holding the headers in place with blue tape, easily one of my favorite "tools".
Now on to the breadboard! I connected the four joystick directions to inputs 0-3 on the bluetooth module. These correspond to Up, Down, Left, Right, respectively. Then I pushed the pair button, and my laptop saw the device and immediately connected to it. Then, I fired up a text editor and created a grid of characters. If everything is working, I should be able to move the cursor around in this grid with the joystick. Success!