Well, I remember in Call to Power you could use corporations to advertise products in a city, creating all sorts of unfulfillable materialistic desires that increase unhappiness.
Maybe you can invest in advertising in cities that have your monopoly/trade HQ resource and have a similar effect (more unhappiness, more demand for your products and therefore more yield from trade routes), amid other effects. That sounds fun, and maybe there's a way to implement it without requiring a unit to make it AI friendly.
Another effect to simulate would be making the best people in other cities want to work for your corporation, creating a brain drain effect as they focus their talent on your projects (drain science from that city to your civ, maybe via trade routes).
Another effect to consider: if your trade routes are disrupted, your products banned or whatever, you can create a special "smuggling" trade route to a city to smuggle and distribute contraband. This increases disorder in a city and creates an underground economy, sapping gold from it as well. That sounds extra fun.
If a civ closes off their cities to your monopoly/corporation's nefarious products, you can demand that they open their markets or else. Think the British/Chinese and the Opium Wars. You can use smugglers and advertising to stimulate demand (disorder, unhappiness, crime) until they're forced to open up again. You can also close off another civ's corporations/start a trade war and possibly provoke a shooting war with them, like the US did with Japan (allegedly).
That way corporations would also have a role to play in city flipping, diplomacy or just causing general problems in another civ during peacetime.
Well. So much for not having any input.