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I’ve been looking at your trend and to me the controller tuning is clearly too aggressive for this process. Note the “porpoising” action of the CV as it tries to get PV to approach SP following the SP step-change. Only two types of controller action can cause this to occur: proportional, or derivative.
Porpoising is when an oscillation occurs prior to PV crossing setpoint, which explains why integral action cannot be to blame here: the only way a loop oscillation can occur is when the final control element oscillates as well (i.e. changes direction), and since integral action will never change direction until PV crosses SP, oscillations that occur on one side of SP cannot be caused by integral action.
Looking at the phase shift between PV and output during the oscillations, it appears the output peaks may slightly lead the PV peaks, but only slightly. This suggests that proportional is the action that is too aggressive.
The large proportional gain is multiplying the error significantly, remember the proportional component is CV=E x Kp, which explains why as soon as you introduce an error you see the valve “smacking” fully open and fully closed.
A proper model will help you get this loop under control.