Another war booty thing that Germans might've found useful: a good HMG made in Belgium. They don't have one in service (bar the token examples) until some time in mid-1943, so the 13.2mm HMGs can help them to bridge the gap. Up to the point that the MG 131 is removed from the production plans.
A HMG has it's role as a defensive weapon for the bombers, and might've fit in the places that a 20mm cannon will not. The Belgian gun was also much more potent than the German type, so letting it fly with API ammo will mean easier penetration of the pilot and fuel protection of the enemy aircraft.
French are also making the HMG in that calibre (even though their are not as advanced as the Belgian types), meaning that there is a lot of potential for the good numerical supply both of the guns and ammo.
The Belgian 13.2mm Browning seems to a unicorn with golden horn.
The French 13.2mm round is basically a US .50 cal that that traded velocity for bullet weight. Not to start with but that is the way it wound up.
And let's remember that the Browning gun, at least the large ones, didn't like synchronization that much. American, Italian, Japanese they all had a major change in rate of fire.
For a captured design to be useful for the Germans they needed not only the design, they needed the manufacturing facilities. And they needed a manufacturing facility that could make the desired gun in large numbers. AND not impact other weapon deliverers also desired by the Germans, pistols and/or bolt action rifles.
I will also note that machinery that can make pistols may not be suitable for making large machine guns. Or even machinery that can make bolt action rifles and/or shotguns.
The German MG 131 is actually a rather good gun for it's
intended purpose. Which was to fit (mostly) into places where the MG 17 fit. It offered just over twice the power for only 50% more weight.
The Big Browning (either US or Belgian) was longer, wider, taller and a lot heavier. It was more powerful
unless that synchronization problem crops up. The MG 131 used electric priming and while the rate of fire slowed down it was more on the order of 10% rather than the 30% (or more) that the Browning guns suffered. Turns out that a MG 131 firing at just over 800rpm is within a few percent of the weight of fire of a .50 cal Browning firing at around 560rpm (some synched Browning's didn't hit 500rpm) Except the MG 131 was a lot lighter.
It was still a lot lighter than the Belgian version. The Belgian version fired faster, so the rate of fire is up but the advantage of the Belgian Browning is not quite what it appears.
Now the Germans screwed up many areas of aircraft armament. Like never really getting a twin mount/turret for the MG 131 into wide spread survive. And if they can't get a twin MG 131 into service the chances of a twin 13.2 Belgian Browning is about zero.
Sticking as single big Browning into a turret or hand mounting where a 20mm MG 151 won't fit instead of am MG 131 isn't really going to buy much.