My point is that a production line that is cranking out a few dozen guns a month might not be of interest to the Germans, they need hundreds or thousands of guns per month.
Germans can start increasing the production in Belgium, include the production line(s) in France, and make the guns in Germany proper. Or, don't make a single HMG in Germany proper, make more of the 20mm weapons there instead.
The German need for the HMGs was far lower than their need for the 20mm cannons, that were favored weapons in all 3 Wehrmacht arms.
They had also been working on their own MG 131 since at least 1938 and they were specifying it in number of aircraft in 1939-40 (shows up in some of the FW 187 specs).
Now it doesn't seem to have gone quite as planned and the MG 131 seems to have started trickling into Luftwaffe service in the spring of 1941 in bombers. For a gun that was supposed to have been fitted into cowl mounts that took quite a while to happen.
By the same token, the Belgian Browning could've been trickling on the German bombers from 1941 on.
The MG 131 shows in the 1942 drawings for the Fw 187. It shows in the very early drawings for the Fw 190, as an wing-mounted synchronised weapon, but that place was quickly taken by a more substantial weapon, as one can expect.
Now the American .50 cal was certainly not in great shape in 1940 or 41 but for some reason everybody assumes that the Belgian version was ready go, reliable and durable and available in large numbers in the summer of 1940.
It took ten factories in the US to satisfy the demand for .50 cal guns at it's peak during the war.
German position on HMGs is very much removed from the US position on HMGs. They can opt not to make a single HMG and their combat effectiveness is not changed by a single iota.
Nobody said that the Belgian Browning was available in large number in the summer of 1940. Seems like it was more ready to go than the MG 131, though, and it was certainly more powerful of the two.
The Germans don't seem to have had FN build very many (or any?) BARs and FN had the tooling to make BARs and even offered a quick change barrel version for sale in 1930s.
BAR probably didn't offer anything over the MG 34, 42, ZB 26 and ZB 39, nor it was filling a niche that was present. Perhaps supplying the raw materials for these MGs was seen as the more prudent move than supplying the raw materials for the BAR?
OTOH, Germans sometimes (many times) didn't seem interested in the foreign designs as-is (bar what they found in the warehouses) until the proverbial hit the fan? Even their usage of the Czech factories was
meh, IMO, until 1942, eg. the capable sFH 37(t) went nowhere.
I don't have reasons. But there was a lot of tooling in the FN factory for all sorts of weapons. The question is how much tooling they actually had for the 13.2mm machine gun and if the 13.2mm machine gun was actually a finished ready to go project. Or was it pretty much in the same state the US .50 was? unreliable in high altitude cold weather, feed problems while pulling Gs and so on. The Swedes and Finns had a while to sort out the problems.
Again, see this from the German perspective. They also have months to test and, need be, debug the Belgian Browning, and it is not a weapon that will make or break their warfare anyway.