Desoldered the codec chip and all its corresponding capacitors.

Heating problem went away and board still works, but no I2S audio.

Made extra sure about the I2S connections by jumpering them directly to the correct BlueGiga chip pins. Still no I2S audio.

Did some continuity testing, and the SCK pin on the BlueGiga chip has continuity with the 3v rail. Maybe that's the root of it.

I'm back down to analog audio now.