Why was the Akron’s spy basket installation so remarkably unstable?
Very interesting bit of physics going on here with the cable car swinging "up to the equator" of the airship.  Perhaps something similar to what happened with the "Galloping Gertie" Tacoma Narrows suspension bridge in 1940.

The suspension cable could be hundreds of feet long.  In a cross wind, if the mass to surface area (wind pressure) of the airship and the cable car did not match, the cable car would swing to the side and then back.  If the wind and the swing motion became harmonicly re-inforceing, each swing becomes a little higher, eventually reaching a full 180 degree sweep.

The elasticity of a cable several hundred feet long may have contributed, stretching a little at the bottom of the arc and contracting a little at the top of the swing, helping propell the object back in the other direction.

Another factor have been a very slight difference in airspeed as the car swung away from the wind, pulling the airship with it, as compared with the return swing back into the wind.  Each swing adds more energy to the system.

Without "dampers", the cable and car are wildly out of control.  The fix, amazingly, may have been to fashion adjustable "sails" for the cable car. Some wind tunnel testing may have been
helpful as well.