When you start getting into hydrograph routing (flow rate vs time), you are leaving the GIS realm and get into specialized hydraulic modeling software. HEC-RAS is great for open channels, and can handle pipe flow, but needs hydrology (how much runoff are you getting) input (often from HEC-HMS). The mix of hydrologic and hydraulic modeling required to (relatively accurately) do urban stormwater flood modeling is a really narrow niche field. The models all have a bunch of calculation options and methodologies, various input parameters to select from and assumptions to make. Plus, most of the commercial software is pretty expensive.
Don't be offended by this, but rather than look for another piece of software, I suggest you look for another person to take this off your desk. For perspective, if a civil engineer with a CAD background said he was going to create some massive multi-user spatial database from scratch and wanted to know which software to use, what would you tell them?