The GSM full rate speech codec operates at 13 kbits/s and uses a Regular Pulse Excited (RPE) codec - This means that the microphone and speech detection in GSM is optimised for transmission across a Time Division Multiplexed 'digital' channel that is then modulated across the air interface using GMSK, a continuous-phase frequency-shift keying modulation scheme.
Noises other than the 'average' speech pattern are heavily distorted (or suppressed) - For instance DTMF (tones) are not well received on a device, and must be transmitted by the network core, but tones designed for the hearing impaired work well. Voice is shaped (filtered) on entry to the codec (microphone design) for best codec detection and reproduction at the other end.
In summary - It is not possible to 're-modulate' across the GSM system, because the entry point is not a radio (air interface), or even access the GSM digital frame. Your only access for a voice call is the GSM codec which is expecting a voice in a confined audio spectrum.
However - If you can create a voice like waveform for the GSM codec, then you may have a way forward