Looking back at my own account, I forgot to close out this thread.
I snapped the point to the line, the hard way. I enumerated all vertices in the line, looking for the closest vertex. Then, I moved the point to that vertex.
This will not always give you a good answer, since this will actually snap to the closest vertex, not the closest point along the line. An improvement is to first densify the line (adding points along the line at a given distance interval), which is part of the JS API.
Ultimately, most people would just use the SnappingManager with the editor toolbar, but this was a special situation.