A little digging around Foobar2000's options, and it looks like I've got gapless working on the external player.

Here's what I did in case anyone else is curious:
File > Preferences > Playback > Output > Increase the buffer length. 5k ms seems too low and still prone to gaps, but 10k ms has eliminated them entirely.
The obvious and significant downside is that the player needs to pre-buffer a bit, and re-pre-buffer(!) it if you change tracks, but if that doesn't bother you, it seems to work pretty well.