Bluetooth mic input is typically sampled at 8 kHz. The weighting filters in the Meter and the bandpass filters in the Octave analyzer are designed to operate at multiples of 44.1 kHz and 48 kHz, which would explain the "Unsupported Sample Rate" message.
It seems that some time ago Apple changed the behavior of Bluetooth audio in iOS (at least in the Measurement Mode that SoundMeter uses). Since SoundMeter requires both input and output from the audio system, Bluetooth won't work unless you are using a Bluetooth headset (or hearing aids, in your case) with both input and output. Unfortunately, if you use a headset for input and output, then the sample rate will be restricted to 8 kHz (for both input and output).
This is a change from how Bluetooth used to work on iOS, when it was possible to use the built-in mic with Bluetooth output.
I wish I had better news.