I might be biased towards low-level reasons, but I bet your main problem is filtering.
You don’t have any of the harmonic notches configured. Start here (Managing Gyro Noise with the Dynamic Harmonic Notch Filters — Copter documentation) and aim at INS_HNTCH_MODE = 3, which is ESC telemetry based filtering, for which you already have the sources working (nice ESC entries in the logs). This will get rid of 200ish Hz noise peak in the filtered data. Here is your current noise profile for reference (a web tool for this: ArduPilot Filter Review).
I blindly suggest the following parameters, which are quite typical:
INS_HNTCH_ENABLE = 1: this is where you start, and you have to reload parameters (or reboot) to get the rest of the parameters.
INS_HNTCH_ATT = 35: the default-ish attenuation which usually works well; this can safely go up to 40.
INS_HNTCH_FREQ = 30 and INS_HNTCH_BW = 3: here, the first is the minimum prop rotation frequency you want to cover with the filter, and although your hover is at about 200 Hz, you want lower values to be covered so that you can descend nicely, and especially you likely want it to be below 50 to prevent frame resonance excitation in that regime. For the second, what matters is the ratio between the first and the second, and for four nicely reported RPMs you want this large enough (way bigger than the default ratio of 2) to reduce negative implication of filters to the overall response.
INS_HNTCH_FM_RAT = 1: I prefer to keep this at default in ESC-based filtering settings, as otherwise the semantics of the two values above become quite complicated to explain.
INS_HNTCH_HMNCS = 7 as you have 1st, 2nd and 3rd harmonics visible in the noise profile. If your FC complains, try decreasing to 3.
INS_HNTCH_MODE = 3 for the filtering based on ESC telemetry as we want.
INS_HNTCH_OPTS = 6 for bits “multi-source” (absolutely mandatory here) and “update at loop rate” (normally works well).
INS_HNTCH_REF = 1: don’t forget to set this, or your filters won’t hear your ESCs at all.
You also have what seems to be two frame resonances at 57 (roll and yaw) and 65 (pitch) Hz. The first recommendation is to check whether you can eliminate these resonances by checking what is loose on the machine and fixing that. In my experience, carbon frames usually resonate at higher frequencies, so this might be something else.
If this is checked and nothing serious was found, I would configure the second notch filter to 61 Hz, maybe as a double notch, to cover those. If you do it successfully, you will see a sharp decrease in post-filter noise, and likely in pre-filter noise too. Here my suggestions are:
INS_HNTC2_ENABLE = 1
INS_HNTC2_ATT = 25 by the visual height of the peak
INS_HNTC2_FREQ = 61: the mean of the two resonances
INS_HNTC2_BW = 30: we want this quite wide
INS_HNTC2_HMNCS = 1: keep only the 1st harmonic
INS_HNTC2_FM_RAT = 1: this does not matter, keep at default
INS_HNTC2_MODE = 0: fixed notch
INS_HNTC2_REF = 0: is unused but kind of reinforces the “fixedness”
INS_HNTC2_OPTS = 1: for double notch, which is at least worth trying here.
I strongly suggest a test flight (preferrably in Stabilize: flying managed modes on an untuned machine is not the best idea) to see if all these changes took effect. I think you will immediately hear the difference in the sound of the propellers. Once happy, proceed below.
Because of all that unsuppressed noise, you apparently had to set up INS_GYRO_FILTER to about 60 and ATC_RAT_*_FLT* to about 32, which is, arguably, quite too low for a 6" quad that flies at throttle of 0.23. I would aim here for at least twice as much as that - once the noise is masked, of course. My 5" machine, which is probably quite similar, has the primary gyro filter set to 150 and rate filters set to 100, and feels quite well.
Once you raise these to something comfortable, test that in another flight, and don’t get fly-aways or nasty sounds, you are welcome to re-run autotune and hopefully get a better tuning. Roll/pitch accels closer to 350K are a good indication, my 5" sluggy slug has 500K+. You may also probably want to tune Yaw(D) if you do active flying (I’ve seen some sharp yaw changes in the logs), for which don’t forget to un-zero the corresponding ATC_RAT_YAW_FLTD.