This signal uses Offset QPSK with a half sine pulse shaping filter for the purpose of getting a constant signal envelop. Does anyone know the exact bandwidth of this signal after being pulse shaped, i.e. the nul to nul bandwidth of the main lobe?
I found this link that details the spectrum for 802.15.4: https://www.semanticscholar.org/paper/IEEE-802.15.4-PHY-analysis%3A-Power-spectrum-and-Gupta-Wilson/2892da39a7ecec1945595d46950438a5bb8777af
And pasted the plot below.
This is consistent with what I would expect for a half-sine pulse-shaped pulse for a symbol rate of 1 MHz (and that would apply to BPSK, QPSK, OQPSK etc where the power spectrum for a random data sequence is the Fourier Tranform squared of the base pulse that is transmitted.). In Offset-QPSK the I and Q channels can be viewed as two independent (in data content) BPSK transmissions, in quadrature, offset in time by half a symbol. Thus each produces an independent but identical power spectrum that adds in power.
Below I compare the Fourier Transforms for a rectangular pulse of 1 us duration vs the half-sine windowed pulse of the same duration. I normalized the power of the two to be the same. Although the pulse shape increases the width of the main lobe, there is a clear spectral improvement by doing the half-sine pulse shape as demonstrated in this plot (Notably, the power spectrum of the rectangular pulse rolls of at a rate of $1/f^2$, or -20dB/decade, while the power spectrum of the half-sine pulse rolls of at a rate of $1/f^4$ in spectrum).
This clearly is not as good as that which can be achieved with more spectral efficient pulse shaping such as raised-cosine shapes but offers the significant advantage to low power systems of being a constant envelope modulation when used in an O-QPSK modulation. In this case we are able to drive the power amplifier (PA, which typically dominates power dissipation in the transceiver) into saturation without introducing AM/PM distortion. In saturation the PA can operate at a much higher efficiency (Tx power radiated at a given DC power in).
I think that your confusion comes from your definition of bandwidth as the mainlobe width of the pulse. In this case it's always the rectangular pulse that wins because it has the narrowest mainlobe. You get the same with windowing for spectral estimation: a rectangular window has the best resolution due to its narrow mainlobe, but its sidelobes are large and decay at a slow rate, so you'll have a lot of leakage.
Coming back to pulse shapes, the rectangular pulse has very slowly decaying sidelobes, which makes its effective bandwidth larger than the bandwidth of methods that use smoother pulses. You can better look at the $99$% power-containment bandwidth, and with that definition of bandwidth the rectangular pulse will come out much worse than the half-sinusoidal pulse used in MSK.