Apply gap filling CA all around the Spruce firmly gluing it in place.

Bend a "whisker spring" out of .015" spring steel music wire. Each "arm" of the spring should be about one(1) inch long.
I tried to show a 2 view of the spring in the 2 pictures below. The idea is to only have a very small contact area of the spring to the door.

Drill a 1/16" hole in the spruce so the spring will contact the doors between the hinges.

Install the spring in this hole with a #2 x 3/8" selftapping screw.

Drill a second hole and install another screw to keep the spring from rotating.

Bend and adjust the spring tension as necessary to keep the door snapping shut when you push on the door lightly with your finger