Linux – A2DP sink without pulseaudio

Without knowing details about the mentioned Bluetooth Music Player, it seems to use simple Bluetooth data connection, otherwise you would not need to install a client on playing/sending device.

To stream audio from microphone to another device, you can record it on your sending device and send it to the receiving device. You will need to implement a protocol for that purpose.
OR
You can implement an alternative A2DP sink service. This is, what the sink is: a device with a Bluetooth Protocol Stack with an implementation of A2DP Sink.

Edit:
For the case you detailed by your comments, the sending device should be left as-is, without installing any app. That implicitly means that your solution must make use of out-of-the-box Bluetooth functionality of that Android device.
What you can use here is therefor limited to those profiles that Android typically support, which is HSP, HFP and A2DP. Since you obviously want to stream music, A2DP would be your choice.
On the device supposed to receive the audio stream and do the playback, you have to implement a service providing the A2DP sink as an self implemented BluetoothService opening a BluetoothServerSocket on RFCOMM as described in Android documentation.

You will have to spend much effort implementing this, and I am not sure if you will need a license for this.

So the above answer isn't totally correct.

The following is how it breaks down:

HAL, is the hardware abstraction layer that implements the actual Bluetooth state machines in c/cpp code, as such it controls the various state machines for A2dp, HFP, GATT, SPP, AVRCP, etc. services. Each of these services also reference SMP and ATT files for controlling the actual Bluetooth server or client databases, and there security.

HCI, is where the actual work gets done. the HAL doesn't really do anything, it assembles the complex data messages that get sent along a tty serial (either spi, or UART) to an inter-network connected chip on the PCBA via the methods used in the HCI layer, which can be found in the "BTE" layer in /external/bluetooth/bluedroid/ directory of an android compiling trunk from AOSP 4.2.2 to current. - currently there are several manufacturers of these chips but they are mostly all Broadcom based ic's packaged in a dual, or triple radio package that contains a wifi, Bluetooth 4.0 Smart, and Bluetooth 4.0 radio.

It is possible to do what you are trying to do, but you would need to include hardware.so, and bluetooth_jni.so into an NDK/JNI package/project that goes with your apps, and registers via the calls from the .cpp files for each of the Bluetooth services found in "Packages/apps/Bluetooth/jni", you would then handle the registration in your NDK library of 'com_android_bluetooth_a2dp.cpp', and 'com_android_bluetooth_avrcp.cpp', as their appropriately typed objects.

The other issue is, you will need to implement your own custom A2DP stack, as the Android Bluedroid stack only has bit's and piece's of the Sink role implemented in the frameworks while the A2DP role has a full implementation of Source role. Additionally, depending what you actually intend to do with your Bluetooth A2DP sink implementation you will need to implement AVRCP as well - as per the Bluetooth SIG (special interest group), there are inter-connectivity requirements between Bluetooth devices that will lead to major issues if you implement sink role, without AVRCP "remote control target device" and "remote control control device", as the sink role ATT commands from Bluetooth over A2DP (or any Bluetooth service/profile) execute certain handshakes during the service discovery process, when the associated gateway ( the connecting device ), executes a capabilities request the A2DP service is expected to implement i/o capabilities for Start Stop commands, and possibly skip/track advance commands.

Additional to all of this, when implementing A2DP you will need to choose whether you will be handling PCM streams or AAC streams. If you are handling AAC streams (or DRM protected PCM streams for that matter, which anything like Pandora, spotify, etc. uses), you need to implement the SBC Encoder or Decoder appropriate to you implementation, else all you will have is a bunch of encrypted data. Also, be sure to implement the bitrate at the appropriate speed for your devices AudioManager implementation, some phones use 48,000hZ and some us 44,100hZ, this is important if you want high quality audio as generally most PCM A2DP implementations that are utilizing Surround Sound 7.1+ will require 48,000hZ as well as AAC encoding/decoding.

I hope this provides you with some insight.