After much head-scratching and experimentation, I finally figured out that the problem was on the Flash side of the equation. I'm playing back the videos with an FLVPlayback component inside Flash, but loading files with the PHP method I was trying doesn't work with the FLVPlayback component. It does work with the less robust Video component, but even using that component, the FLV won't progressively load into the component like it does when you load the file directly from Flash, which is causing problems with creating user-friendly loading bars.
To say I'm disappointed with Flash would be an understatement. Nevertheless, the input I received from this board was much appreciated. And hopefully the findings in this thread will save someone else similar aggravation.