I've not seen Flesh-footed Shearwater, but I'm reasonably familiar with the extremely similar Sooty Shearwater. I've never seen Sooty looking so heavy-bodied as this bird. The bulk looks to me to be a better fit for Westland Petrel.
I don't have a lot of experience, but I would think the length of the bill would rule out most petrels. The dark bill tip would seem to favour Westland Petrel. Just my two cents until someone expert comes along.