I have a U360 Oval joystick
The "Ikari 12-way Rotary Upgrade for Servostik, J-Stik" kit is only for microswitch sticks.
As you observed, the U360 joystick won't work without the magnet and sensor.
AFAIK the only U360 rotary kit currently available is the
UltraStik 360 Rotary Joystick Upgrade,
but this is a spinner-style "non-clicky" optical rotary kit, not an Ikari-style 12-position "clicky" mechanical rotary kit.
There are 12 position rotary switches with a center pass-thru hole instead of a shaft that might be used to make a mechanical rotary U360 mod, but that would almost certainly involve fabricating a custom shaft.
- The fact that Andy doesn't offer a mod kit like this tells me that it is very difficult to design and/or source parts for it.

Scott