You have the popping in terms of the timing, but not the mid-flame flare.
I'm thinking something like this (exaggerated some from what I would probably do), just ~one frame flashes of this on each "pop":
You are also scaling the glow of the flame during the pops, but I don't see anything when the flame comes on full. I'd make the glow about 50% wider and narrow to where it is now over 1-2 seconds, and the flame itself be slightly narrower and grow over the same period. Similarly the "lines" on the outer edge should have a slightly wider angle at the start and narrow at the same time, perhaps even add some randomness to the direction.
And yes, combining Unity a Particle System and an Animator is a pain to preview since there's no clean way to keep them in sync. You can kind of hack it by having the animator toggle the particle system component on and off so you can at least loop in play mode.