Fixed it. It turned out to be a configuration error on my part.
I have it working on three different platforms: an ESP13, a WeMos, and now a NodeMCU v1.0.
I like a lot of things about the WeMos. However, even without an RFM69HW, I had doubts as to whether its tiny voltage regulator was truly adequate for supplying the currents needed by the ESP8266, and adding the RFM69HW seemed to push it over the edge to failure. So I replaced its voltage regulator with an LD1117-3.3 that I had laying around, and now it seems solid.
[Edit: "Witty Cloud," which is yet another platform, comes with a much beefier voltage regulator than the WeMos, and so it may be worth considering as well. ]