When placed on the ground (in this case dense turf, but also happens with the turf removed) in a shipwrecked world, the idol becomes a sunken object and can't be picked back up. You can cast your fishing rod at it, but gets cancelled immediately. I seem to be unable to retrieve the idol again. This bug goes away if you remove the idols via console and retry.
Be in RoG. Have a lost idol in your inventory. Take the Seaworthy to Shipwrecked. Drop the lost idol anywhere on the ground nearby.