Other than direct radiation, how can RF interference transfer between stacked PCBs?
Electric fields will couple everywhere, everywhere water can flow. If there are holes, even if not line-of-sight, the electric flux will enter and upset the nodes behind the shield.

The coupling of electric fields, ABOVE A PLANE, drops off as the CUBE of the distance. (thanks to Howard Johnson for writing of this) Thus Efield coupling through the air, in the butterfly setup with the spurs adequately low, becomes unacceptable and desenses when the 3 boards are all folded up.

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Do you count Efield (such as parallel-plate or wire-wire or wire-plate) coupling as "direct radiation"? 

This is all near-field, where the E and H modes are accurately modeled separately.

Thus if radiation of any form is not the path, you are left with wires.

But since "closeness" seems the cause, then Efields or Hfields may be the culprit.

Can you make sketches of where these are possible?
and can you identify the signal injection levels (such as -140 or -120 dBm) that cause desense?