Bake blender particles for export (to Unity)

I’m trying to export a particle system to Unity to animate a trail renderer.
I’ve keyframed stuff like this by hand before, which export just fine. Now I’ve set a particle system to render objects (in this case, an empty), and I assumed baking them would allow me to export these, but I just don’t know enough about Blender to figure this out.
Here’s an example of what I’m hoping to achieve, it’s a keyframed empty, with a trail renderer on it in Unity (sorry for the terrible framerate, Unity’s acting up)

I’m testing this to see if I can get a nice looping firefly animation going. Since particles in Unity can’t have a trail renderer on them, I’m testing this way. So I suppose what I need to know is, can I record a particle system with physics to actions in Blender?