I agree this seems weird. You did not mention the type of power supply used nor its current, but perhaps its a good low noise supply used with your other rigs?
Of course to make more power, which is typical at lower frequencies, it must draw more DC current (and dissipate a bit less in heat perhaps). I have no issues using my uBITX on 160m, into a dummy load or my decently matched vertical with a 3 amp supply made for ham radio use.
I would definitely checking the wiring for both RF and audio interfaces. Make sure you don't have the ground connection on the board connected to the center pin of the antenna connector. Please note there are 2 power supply connections - one is segregated to feed only the PAs - of course both need to be connected to your power supply. I would look to see that audio ground is going to the expected place on the headphone jack. Use very light audio gain for your testing, maybe 5 or 10% of the volume control should produce headphone audio of the sidetone.
The only other thing I can think of for buzzing is to inspect for any broken or missing capacitors.
As I write doing my own debug - its part of the journey.