Actually FN used 3 waffenamts throughout their occupation and all 3 are known for barrel production/acceptance. (e/613 e/103 and e/140)
MO and FN had a close relationship because they were sister firms, - DWM at one time owned both firms until 1918, where FN was outright stolen from them by the Belgian government in a flagrant violation of commercial law. DWM/Luwig Loewe was not a government owned firm in 1919, yet they were stripped of ownership all the same.. the nazis in an unusual act, decided to give FN back to DWM (and they put DWM/MO managers at FN- though still heavily under military oversight).
Of course this was more symbolic in nature, as "private" corporations at the time were strictly regulated, from materials, labor, to products made- as well as profit margins, currency laws etc.. making them essentially private owned but in every way government managed.
It was unusual though for the time, as “generally” firms like FN would have been absorbed by the Reichswerke (through a subsidiary like Steyr or Rheinmetall) or some government owned firm like Gustloff Werke/Trust. Most likely it was the people behind DWM that made this happen- the CEO of DWM was a big-nazi dirtbag, with lots of connections (x-wife was married to “Gerbils” the propaganda minister)- and of course the number one best friend of all socialist governments (from Hitler to obama), the banks, were the REAL owners of LL/DWM/MO 1890s-1945.
All this will be covered in the next MRJ -