Mansfield puts so much narrative weight on the fly because the fly's situation describes exactly how the boss feels about his situation. The fly is him. He is enacting on the fly the sadistic treatment he feels British society is doling out to him.
The drops of ink that weigh down the fly are like the blows the boss has received. The worst was losing his beloved son in World War I. But added blows are the way society tells him (and by extension all the many other people in his situation of having lost a child in the war) that he must keep a stiff upper lip, stay optimistic, and keep on trying. This cultural imperative is cruel to him: he wants to grieve; he wants to rest; his heart is breaking. Yet as he admonishes the helpless fly, he too must keep on getting up after being staggered with weights he can't bear. The latest is the way Mr. Woodifield comes out of the blue and rips the scab off the boss's wound with his talk of visiting battlefield graves. Mr. Woodifield's visit is like one of the big drops of ink on the fly: it staggers the boss because it makes it impossible to forget.
Mr. Woodifield's visit is directly related to the boss's killing of the fly: Mr. Woodifield has put the boss into intense pain, and the only way the boss can relieve it is to do to another innocent creature what is being done to him.