I'm not convinced it won't, just not sure it was the place to look. A couple of other articles (like this one) appear to suggest that there's a "B-1 in lieu of H-1B" exception for when you're being paid entirely by a foreign company for work being done in the USA, which nails down my situation pretty much exactly.

So yeah, actually, it looks like a B-1 is the right answer, even though the State Department website suggested otherwise on first inspection. I'll follow up on that, thanks.