When A Recess Becomes A Hole

I've been working on another Dutch Tool Chest, but this one has some surprises.  So, SURPRISE: this Dutch chest has a drawer!  Shocking, right?  Every good showman and politician knows to cover with hype what is weak or boring.

What isn't hype are the pulls I purchased for this chest.

Tonight I was doing some test fitting.  The pulls are meant to be fully recessed.  The problem is that they are just shy of 3/4" deep--the same depth as the wood.  I'm too cheap to redo all the front parts in 5/4 wood, which leaves me with a hard choice.  Either I get new pulls, or mount these above the drawer front.  Here are a couple photos to illustrate.

 I'm mostly posting these so I have place to refer to them as I try to decide which way to go.

  

I"m tempted to go ahead and risk the tight tolerances and mount the pulls anyway.  The chief problem with that approach is the screw length.  The screws have to be filed down to fit the remaining wood.