Relevant discussion from another community: https://www.eevblog.com/forum/reviews/any-opinions-on-the-aixun-t3a/
Buzzing isn't mentioned (or audible in the video), but it might be that build quality varies, or you have a clone or whatever.

Here's the spectrogram (waterfall diagram from left to right) of audio from the video.
The blue and pink confetti is just background noise that can be ignored. Most harmonics are in red, with the 1.5k and 1.66k harmonics being most prominent. Probably because they're resonances of the handle or something. It's subtle, but the harmonics seem to swish around a bit, in the bracketed area, distinctive of PWM changing.
So yeah, it's fairly low harmonics (9th and 10th, and you'll still hear the rest; 20th+ would be more practical to filter electronically).
If you want purely electronic solutions, it could indeed be filtered. The impedance is low enough that a catch diode and series inductor could be added, and maybe filter cap too. Downside is, we need Fc << 166Hz. Say 40Hz to avoid cutting into the control loop bandwidth too much; this needs L = \$\frac{(3\,\Omega)}{2 \pi (40 \,\textrm{Hz})}\$ = 11mH, rated for up to 8A. Which will be a fairly large iron-core inductor. (A cap of \$\frac{1}{2 \pi (3\,\Omega) (40 \,\textrm{Hz})}\$ = 1.3mF can additionally be used.)

simulate this circuit – Schematic created using CircuitLab
which would look something like this. Note the diode must be placed adjacent to the switch transistor, and the inductor probably won't fit anywhere, it'll have to be hanging out the back of the base unit or something.
Smaller values may be reasonable (evidently around 500Hz would be the maximum to still get value, given the spectrum), but mind that they may distort the response (power output vs. duty cycle), which may upset the controller as well.
Ideally, the controller would run at much higher frequency (over 20kHz is easily done), with a compact LC filter inside the base. I have no idea if this is at all feasible to implement. You'd at least need the schematics and firmware, and a toolchain to reprogram it.
I mean, even more ideally, the thing just wouldn't buzz at all... Whether that's shoddy construction in the handle, base or both, or poor cabling, or a poor tip (try one of those famous JBC tips you mentioned?), who knows. Also, you might want to check wiring quality in general; with 8A peak flowing here, it would be very easy to skimp on wire size (don't forget they're fond of copper-clad aluminum or steel wire over there..!), not to mention the mains wiring and especially ground connection may be woefully inadequate (i.e., shop for a proper UL-rated unit).
as the tip heats up, there is an audible buzzing noise coming from the handle, the control board, and the power supply.This could be the power supply as well i.e. it might be struggling to deliver the required power (192W times the duty cycle). \$\endgroup\$