Turns out it's a pretty simple mechanism actually... there's a base plate (the "shoe"), and a clamping "thing". The bar that goes up from the "shoe" to the lever is threaded, as you turn the lever it tightens, raising the base plate which squishes the clamping "thing" against the underside of the ways.
See photos (click to embiggen as usual):
Lever in unlocked position:
Lever in locked position (note: more thread sticking out of the top):
Looking at your photo Matt, I think all you need to do is to tighten that threaded bar up so it's just touching the bottom of the shoe when the lever is at the back of the tailstock. I'm going to take a wild guess that it'll be quite stiff, I'm assuming the lump on the bottom is designed to bind slightly in the "shoe" so the lever tends to tighten on the thread, rather than just rotating the whole piece. If not, then I'd tighten it anyway, then wedge a shim in there so that it binds up.