Well, long story short, I bought a passive PoE injector/splitter block off eBay, and it arrived today. Unlike my solution, it had the correct connector on it to use the OEM power supply out in the garage. I'm not sure if that made the difference, or maybe their wiring is somehow better, but it works perfectly. You can buy them here.
I didn't have much hope for the passive solution, so while I was at it, I bought a used DLink DWL-P200 PoE injector/splitter kit. Unlike the passive solution,
UPDATE The DLink DWL-P200 arrived today, and it too works perfectly with the Microcell. My guess is that the DLink is the better solution, since it uses a higher voltage down the Ethernet wiring, with signaling between the two ends to enable the supply voltage, which offers some protection to all the devices involved. It also means that the supply voltage going to the device is better regulated than when the power supply was on the other end of 50 feet of twisted pair. The downside, of course, is that the DLink DWL-P200 is much more expensive than the passive solution. Though I was able to get a used set on eBay for cheap. Oh, and it turns out the DWL-P200 is not 802.3af compliant, since it actually pre-dates the 802.3af spec. Still, the higher voltage feeding into a low voltage regulator is likely to work better over long lines.