XRD: Can stacking faults cause pseudo-centring?
The outcome depends on the typical length between stacking faults and the coherence length of the X-ray beam. 

If the length between stacking faults is large compared to the coherence length, you will get a phenomenon similar to that of twinning (unit cells in same orientation) or of broadened diffraction spots similar to the model of high mosaicity (unit cells in slightly different orientations).

If the length between stacking faults is small compared to the coherence length, you will get a phenomenon similar to partial occupancy of sites in an otherwise perfect crystal (i.e. static disorder - same diffraction pattern with different intensities compared to absence of partially occupied sites).

I found a statement about coherence lengths of synchroton radiation (https://www.researchgate.net/post/What_are_typical_requirements_on_coherence_lengths_for_coherence_experiments_using_3rd_gen_Synchrotron_Light_Sources):


  [Keoki Seu, March 21, 2012] I work at a coherent scattering beamline at the Advanced Light Source. It is an undulator source optimized for coherent flux. At 500 eV, which is usually where we work, coherence lengths are on the order of 6 um vertical and 3-4 um horizontal. We'd love to have larger coherence lengths so we can coherently illuminate larger sample areas, but for the soft x-ray, the values we see here are typical values.


The question was about power patterns rather than single crystal diffraction. 


  [OP] Will the situation described cause the powder X-ray reflection pattern to resemble space group symmetry Fm-3 m, e.g. by integral extinctions of reflections where h + k, h + l or k + l is odd?


I don't know what the typical dimensions of the solids in the powder would be, and I don't know what the coherence length of the Xray radiation used in the experiments are, but if you consider all of these factors, you should be able to say in which regime (i.e. twinning-like or static disorder-like) the experiment at hand is in.