Turns out that VLC example is a bit out of date - see the WebPlugin documentation. I can update the HTML to:
<embed type="application/x-vlc-plugin" pluginspage="http://www.videolan.org"
windowless="True"
width="640"
height="480"
src="rtsp://192.168.0.10/screenmirror"
id="vlc">
</embed>
The key is the "windowless" flag, defined as:
Draw the video on a window-less (non-accelerated) surface and allow styling (CSS overlay, 3D transformations, and much more). Default: false
Once I apply this the VLC UI disappears, Hammer.js seems to work fine and my alert appears when I click the video.