Actually it does! Here's the new project, now not following the weapon in setup mode, but following it in the animation.
I've created two additional bones that will stay on the weapon. when the weapon is present, turn on the transform constraints so that the IK targets will follow them.
In setup mode they can be all set to 0 influence, and then set at 100 in animate mode.
Make sure that when you create the transform constraint for the hand ik, you move the ik AFTER the transform constraint in the tree order, or it won't work.
I had auto-key turned off so the animation was actually empty, project updated.