The P360's used to be a switch replacement option, but now they are their own joystick, as you have already guessed.
These bases are not made by IL and will not accept a 10mm shaft (that means Super and Comp shafts are out). It seems Happ "Americanized" the P360 and designed it with a 3/8" (9.5mm) shaft. If you go with a P360, you are pretty much stuck with the stock shaft.