I think it's important to remember in this case that The Sandman was originally published in monthly issues; they were connected by overarching stories, but weren't collected in volumes. Thus I replace the question "Does part four of Doll's House add to the narrative of the volume?" with the question "Why was the story 'Men of Good Fortune' collected in The Doll's House?".
The answer appears to be [somewhat trivial]: Morpheus' most recent meeting with Robert Gadling takes place roughly at the same time as the main story arc of Doll's House.
It was important to have "Men of Good Fortune" appear exactly at this point, because this is when it is set chronologically.
Assuming you read the monthly issues, source of possible confusion becomes clear: Gaiman says that the end of the issue #12 ("Playing House") should have had "Next: The Prior Engagement" written in the end. Editor or typer took it out from the issue, since the following issue (#13, "Men of Good Fortune") does not have that title. My digital trade paperback has the notice:
Here's the maths: Robert and Morpheus met in the year of our Lord 1389, and agreed to meet once in a hundred years:
Thus we know that they must have met in 1989. We can compare that using the dates from the prologue of Doll's House, where Morpheus, immediately after the events on the Skerry, goes to the Threshold, Desire's realm, and says Unity was raped 50 years before that (i.e. it should be 1939):
Going back to Preludes and Nocturnes, however, it all gets confused: we can conveniently see that Unity was not raped in 1939, but in 1932:
If we change "years" to "months", this panel fits into the timeline.
I asked Neil Gaiman on Twitter about it:
Gallifreyan: Hi there! A question about chronology in The Sandman: is this panel's timing wrong? Shouldn't Unity have been raped in 1939?
Neil Gaiman: I think Morpheus is approximating. And probably should have rounded up, not down. But he was in a glass prison while it was all happening.
So it appears that Shokhet was right again saying that the number 50 does not have to be precise; I was right too, saying that 57 rounds up to 60 instead of 50.
And it makes sense for the events of Doll's House to take place in 1989. Consider this panel from The Kindly Ones:
The solicitor says Unity awoke in 1988, same time Alex Burgess fell asleep, i.e. when Morpheus broke out and put him in a nightmare. Thus it is chronologically more probable for the events of Doll's House to be set in 1989 (i.e. Preludes and Nocturnes ended somewhere between 1988 and 1989). Either Morpheus doesn't know that 57 should be rounded up to 60, or that panel from Preludes and Nocturnes must be confusing something.
The Sandman Companion by Hy Bender (page 44) raises an interesting point here: during his encounter with Brute and Glob (in "Playing House"), Morpheus tells Hippolyta Hall that he has "a prior engagement".
The next story after that is "Men of Good Fortune". Thus, Bender makes the conclusion that the prior engagement is Morpheus' meeting with Robert Gadling.
This is backed by Gaiman's words on those pages:
N.G: [...] The prior engagement referred to, of course, was the Sandman's centennial appointment with Hob Gadling in 1989; and it was genuinely important for Dream not to miss it, because it was the meeting in which he affirmed their friendship.
The Sandman Companion, chapter 4, page 54.
Also, in the same chapter 4 of The Sandman Companion, Gaiman asserts multiple times that he intended for their meetings to take place in 1989.
@Shokhet is also right to assert that the comics were supposed to be set in the year they were published, and the same chapter for the companion confirms that Gaiman was writing them in '89.
One could, however, ask: "Why not collect the story in some other volume, like, say, Fables and Reflections?". And the answer is - it'd be too late and crammed. Hob Gadling's story must be told before further volumes, and though dates may be fuzzy, most evidence shows that it fits into the chronology of Doll's House best.