This is the simplified explanation for a possible cause:
When a string is played normally, there are 2 nodes - one at the bridge and one at the other end of the string at the nut or wherever you are fretting the string. The maximum deflection in the vibrating string is exactly halfway between the nodes. For an open string, that is over the 12th fret. There is no string deflection at a node.
Magnetic pickups generate their signal by the movement of ferrous metal through the magnetic field generating an electrical impulse in the coil. The more movement (amplitude), the more strength (volume, amplitude) to the signal.
For open strings and normally fretted strings there is some movement in the magnetic field above the pickups to produce a signal. Again, the more movement the stronger the signal. But the opposite is true too. The less movement, the less signal and the less volume you can generate. The extreme case of this is a string that is not vibrating at all: no vibration = no signal generated.
Harmonics add additional nodes to a vibrating string. The harmonic at the 12th fret adds a node at the 12th fret. There are now two (reduced) maximum amplitude areas - one halfway between the nut and the 12th fret and another halfway between the 12th fret and the bridge. But you can visualize that the string still has movement over the pups, so you get a signal.
What happens when a node is created right over a pickup? Since there is no movement at a node, there should be no signal. But just off the node on either side, the string is vibrating - with more amplitude as you get away from the node. The problem is the string is vibrating in the exact opposite wave on dither side of a node. Any signal produced by these small amplitude movements would be very soft to start with and might even get cancelled out due to the opposite direction vibrations.
I'm guessing that the harmonic for the 5th fret is placing a node right over the pickup that is relatively quiet. Time to measure - first from the nut to the place you create the 5th fret harmonic and then from the bridge towards the pups. Is the node on or really near the quiet pup?
bingo.