I came up with a pretty simple workaround for this: I'm now modifying the z-index instead of hiding the modal. Seems to work great.
function closeModal() {
// We can't use fadeOut because it hides the Uploadify Flash button, which causes uploads to fail
$uploadModal.css("z-index", -999);
}
function openModal() {
$uploadModal.css("z-index", 999);
}