After obtaining a fitting for a 1/4" ID tube, I made all necessary connections to the carbonator and set it going. At 60 PSI of CO2, the water was clearly carbon-tasting, but the bubbles came out of solution very quickly. I increased the level to 100 PSI, and most of the bubbles still came out of solution right away, but a few remained. I'm reluctant to pressurize beyond that, but I'll do some research and try to find the typical pressure of carbon dioxide to get a good maintained carbonated water.
Nonetheless, this is a fairly significant success.