I am a bit at odds with the Drilling as it is made today.
The old-time Drilling was a side-by-side shotgun with an additional rifle barrel. But these guns were balanced and handled like shotguns.

Nowadays the Drilling is seen as a rifle, adorned with a biiig germanic scope ("you simply need 50 mm lenses for night hunting", stocked in the worst germanic way (humpback with a lot of drop, a steep "Kaiser grip" etc., fitted with a heavy full-lenght insert barrel etc.
High pressure rifle calibers will add weight.
So the drilling got heavy, in my eyes out of balance, OK for "Hochsitz" shooting, but pretty useless for decent shotgunning - remember that one barrel is always blocked by the rifled insert....

Having a new Drilling built to specifications is expensive.
So my only choice would be a pre-war Drilling, 16 gauge over 7x57R or 8x57IR.
One simply has to "bite the bullet" that 16 gauge shells may be hard to find, and that most pre-war hunting barrels were made for .318 bullets.

Fuhrmann