I have now gotten FL to find and load my plugin by moving it to C:\Program Files\VST3 and manually adding this path to “Plugin search paths” in the plugin manager, just for good measure. (Not sure if that last part is actually needed.)
This is what the list of paths looks like now. Notice that some paths are labeled VST and some are labeled VST3.
There is still a smaller problem, which is that even though the plugin manager lists my plugin, it leaves the “status” and “vendor” fields blank and leaves “effect/synth” as “?”. The plugin runs, so it’s not a critical problem, but I’m still working on it. This doesn’t happen to 3rd party VST2’s, but it does happen to 3rd party VST3’s other than mine, so it could just be a problem with VST3 in FL.