Naturally, just as I posted this, I stumbled on the solution. So, for those who see this and want to know how...
With the CSSImport plugin, I set the ID of my FileChooser to 'chooser'.
In my style sheet I added this:
#chooser::-webkit-file-upload-button
{
visibility: hidden;
}
And that ugly button vanishes. Or, just apply your own styling to it.