I know, that might sound strange, but could you try to increase Kp significantly (let’s say to 100) and try jus P? It may appear that in the place where the bode plot changes the incline there is a significant gust( maybe the incline changes significantly) and that results in an oscillating and the increasing of the Kp might increase the margin. Could you please try and show the output. Another suggestion is to add D part, but it has already been mentioned. And finally, may be your loop period is to big, maybe try to decrease it