Alright, here's the solution at last!
Moving all the glitching objects into a new group IS the answer. However, you must do them all at once. No stopping to test them in a map. I did that with a few objects to make sure they worked. They did. Then I went ahead and did aaaaall the other objects in that group, and when I tested them in the Cape, they didn't show up with the first few objects.
So:
Move objects to a new group.
No testing in between, do them all at once.
Everything should work fine if you do that. This does, irritatingly, mean that you can't ever add to that group again. So if you want to install an object that's just a single something instead of being part of a pack, I recommend waiting until you have a few single objects that would fit in a group together, then make a group for them all at the same time.