Are you actually using the multichannel expander or is it just a Bela Mini? You only mention two channels.
You most definitely don't want to use a linear regulator because that would heat up a lot, 200mA * 7V = 1.4W. And that's unlikely to fix anything, as the issue seems to be about sharing ground between the two.
One option is to try using the audio ground for powering it: plug the - from the power supply into one of the GND pins on the audio side: in the audio input connector, in the headphone connector or in the line out connector. See these three pins highlighted below, pick one of them:
when I power it with the step-down converter I get low bleeps and blops when the Bela is booting.
Before the Bela program starts the codec, you should have the amp muted. Your breakout should have pins for that: use a 10k resistor to set it to muted then connect the mute pin to a GPIO on Bela and unmute it when the program starts.
PS: for the next time, links and product datasheets are more useful than pictures
