Ideally, what I’d like to have is an item in the content browser I can drag into the scene, which brings in the whole object. Essentially a group item/asset which references each of the parts but has them as a whole/merged item.
You could probably export all the pieces in a single FBX file, and then UE4 will probably merge all of this into a single staticmesh upon importing if you configure the options correctly.
That is not exactly what you asked for though. You want a kind of prefab probably, and that doesn’t exist in a simple form. The thing that comes closest is making a Blueprint that has all of these pieces assembled in it. You will need to set this up once but after that you’d be good to go.
Which of the 2 options to take depends on the further use of the asset and your performance requirements. Option 1 is heavier on memory, option 2 could be heavier on rendering.