You could also drill and plug a small void in the tungsten bar to make the fake exact.
My preferred approach would be to cast Tungsten cores into gold bars, so you had only 10% gold, and a millimeter (or so) of real gold on the outside to provide an assayable veneer.
It would probably take some serious technology to get gold to bond to tungsten (maybe the copper over lead bulletmakers could help?). Imagine the shock of dropping a big gold bar, and seeing its gold skin split open, revealing a dull grey interior monolith. It would make a good scene in a movie!
The way I would write this novel, a vault employee who is also a university student accidentally uncovers the secret. The people he tells begin to die, and attempts are made to bump him off “accidentally.” He tells the CFO about the tungsten bars, but learns that the principals already know. He doesn’t learn this immediately, but only after his girlfriend is poisoned (meant to be his drink), they attempt to kill him during a “mugging” etc.
Great potential there for a novel or screenplay.