Turns out planar reflections were a bit too heavy for our game. Still, if others stumble upon this issue with Unreal, I think I may have found a solution/workaround.
If your lights are static, and you don’t want them to appear in your reflections, then uncheck “Visible” in the Rendering section of that light. The light and shadows it produces will still bake into the environment via lightmaps, but it will not appear visible to reflection captures.