Share the LayOut file so we can se exactly what you’ve got going on. It sounds like your model layer is under another layer and the top layer has something on it that occludes the view to the layers below. Or that you are pasting to a layer that is hidden. If it’s the latter you should get a warning as shown below unless you’ve disabled it.
So I’m looking at your LO file. I see the SketchUp viewport is on the top layer called HS elevation symbols which is a weird name for the model viewports but at least it’s at the top of the list so there’s nothing overlaying the viewport when you paste it. Copying and pasting does work but its quite slow due to the complexity of the SketchUp model.
I see in your model that your walls are all loose geometry.
And a ton of unused stuff that could stand to be purged.
There are some rather heavy groups in the model although since you used groups and didn’t name them it’s difficult to tell which ones they are.
Cleaning up the SketchUp model, simplifying the geometry and reducing the size of oversized materials will help to improve performance in Sketchup and in LayOut. Your viewport will copy more quickly if the model is simpler.