Why does increasing the modulation index of GFSK reduces peak power, increases the range & robustness of a bluetooth connection?
Consider case-a with modulation index 0.32 and case-b with moduation index 0.5

In case-a, each bit contributes a phase change of +/-0.32 pi based on the bit being +/-1 and +/-0.5pi in case-b

Since the phase deviation is more in case b (pi) compared to case a (0.64pi) more rapid changes in the waveform are expected and so contribute to higher frequencies in frequency domain.

When same power is used in both cases, the peak power is less in case-b compared to case-a as the contribution of higher frequencies (away from DC) are more in case b compared to case a.

As frequency deviation is also more in case-b compared to case-a, distance between symbols is more in case-b. So case-b can have more range compare to case-a.