So basically this is a "star" network where the data packets are rats. https://en.wikipedia.org/wiki/Star_network
The Exchange is where the interesting problems are, and it also is your bottleneck.
Musing about LOGISTICS.
THE TUNNEL NETWORK
If we're talking about a city of 100,000 rats -- er, households, then you might have a link-and-branch system of ever-widening tunnels until the major conduits empty into the Exchange. I suspect that you would need a reasonably geologically stable city.
You'll need many operators, related to the number of messages per hour. I'd wild-guess that 1 operator could reliably handle 15 messages per hour for eight hours. If each house is expected to send one rat per day, then that's 100,000 rats in 24 hours -- round that up to 4,000 rats per hour. So you'll need 270 operators working every day... maybe 150 of them during the waking hours, and the remainder divided between two night shifts. Or whatever!
For various reasons -- for example, bad handwriting, or failure to pay one's bills -- the Operator may return rats to the sender with a "not delivered" tag. How sophisticated this feedback system becomes depends on the efficiency of the system and the needs of the population. In particular: the more commerce that flows through the network, the more sophisticated the messaging system will become.
Note that if your peak work hours requires 150 workers, then assume you need perhaps 15 supervisors (?), a dozen or so troubleshooters (now there's an interesting plot setup!), and a triumvirate of Bosses to solve really big problems. Plus an above-ground Customer Support Centre with a row of customer service experts whose inter-personal skills rival your Department of Motor Vehicles. And their Supervisor. And a Director (who is too important to have an underground office). So this Central Exchange has airspace and workspace for maybe 200 people per 100,000 population. Maybe. Scale to taste.
THE RAT STACK
You'll need more than one trained rat per household: for example, the house will need one on standby for sending a message, but also the Exchange will want AT LEAST ONE on hand for dispatch. And this is where your city will scale its rates. If you order more than one Central Dispatch Rat due to the heavy volume of orders to your rat-packet-home-shopping-catalog, you'll have a heftier utility bill.
Say the default is one Central Rat. That's 100,000 rats in 100,000 cages in the Exchange, probably sorted into some sort of rodential Dewey Decimal system. Which means the workers in the Exchange are skilled workers; not only do they have to be literate, but they also have to understand the System, and be able to FILE and SEARCH the system with a very low number of errors. Imagine the pain of a mis-filed Rat that gets dispatched, ostensibly to your home, but instead ends up at the Prime Minister's estate. We are not amused!
As an aside, the Exchange will not want to immediately return any rat to its home: if the Exchange's cage for that home is currently empty, they might want to hang on to it. This is where errors will happen, because someone's mis-filed Rat will cause another household's Rat to be held at the Exchange.
Thus you will need a Rat Mitigation Strategy. With a skilled workforce and reasonably efficient checks, this will be minimized, and therefore just a cost of an efficient network. In careless cities that have cut back funding, hilarity may ensue.
Toss in at least some paranoia or threats about a Virus being released into the network, and there will occasionally be political and popular backlash, but if the network is really invaluable, such problems will be paper tigers. In the paper RPG culture, this is what we call a "plot hook".
Assume that occasionally a householder will end up in therapy due to Rodent Anxiety. This is what I call "local color".
Finally we have the fact that such a network, if efficient and effective, will be quite valuable to a city... and therefore it is a single point of failure that needs to protect itself against hostiles. So expect to have security forces -- or just security theatre -- posted there permanently. Perhaps you could place the Exchange close to a military barracks, for a "free" level of security.