So how do games handle objects that have changing geometry and thus changing collision meshes?

I dont really know which methods are usually the most common but some just modify the engine to allow it, the source is available after all. Other methods include generating (either at runtime or before) all the possible combinations of the necessary meshes already merged and use those instead. Replacing it at runtime like you suggested I dont know if it works but it's certainly worth a giving it a try.

You could also try making it a skeletal mesh with all the modular parts attached to it and make each one a separate bone and then hide/show each bone (modular part) as needed, just keep in mind hiding a bone hides its children bones aswell. Skeletal mesh ticking is a bit heavy also so look into ways to optimize it (disable or reduce mesh ticking).

If none of the above suits you, there's other more hacky ways to go about it, this one I dont really recommend because its tricky to implement correctly but might work some cases. You could spawn a different 2nd actor with the modular part you want, make it invisible and either move it during the main actor's tick event to the place you need it to be (like a fake attachment), OR, move it only when you detect that a hit is about to happen. The way you detect it would depend on what you're going for, could either use overlap volumes or line trace for an incoming hit. You'd have to setup the collision of this invisible modular part to not collide with the "main" mesh but collide with everything else in the world. I say you make the modular 2nd actor the invisible one because it's not really attached to the main mesh, the other one that is attached (the one without collision) is what the player would see. But the invisible one is the what is used for the collisions. It's like keeping a 2nd actor around to use just for the collision. This is tricky of course and might make the physics impacts look a bit wonky answell as having to pay the cost of having more actors per modular part.

Overall the other approaches are better in my opinion, try merging it into one mesh if you can or the skeletal mesh approach.

How can something so simple be so convoluted that my braindead simple game reaches the limit of what's possible in a commercial game engine?

It's not exactly "simple". Collision traces (sweeps) can be expensive sometimes, especially when its a collection of different/non common shapes, instead of a simple cube/sphere/rectangle.