Thanks for the reply Michael. MoI definitely requires a new way of thinking, which is very challenging! I didn't know about slicing with a curve like that--very useful!

Here's the spout from the side, where you can get an idea of what I'm talking about:



At the opening, the side surface I'm trying to model is essentially vertical, and very slightly convex (if you looked at it from above). Conversely, where the spout is attached to the wall, the surface has become cylindrical. To me, there doesn't appear to be an obvious base shape to use that could then be carved away with booleans or trimming, which is why I was assembling part by part (after having no luck with sweeps, lofts or networks...). It seemed like such an easy project when I began!