P.S.: Just to be complete, Bake doesn’t work with Hair. BUT:
You can set it to, e.g., Circle, make it a Polygon based object, lose Hair Color, etc., but keep the Curl and other attributes. Which means, you need an new Material, see file.
This allows to bake reflections, and so the example of your file can be rendered. (Perhaps create the Flat objects as discussed above and use this technique for it, I’m not certain how far that carries [refections of reflections], but worth a try; or not at all: )
https://www.amazon.com/clouddrive/share/Bqc4puvKyLU4mn2dpfVHSqRRnckAw2Lpo3dEcGYYncY
I have set up a green-screen in the file. In Photoshop go to Image> Calculations…
Here set up the Red Channel and Inverse, as well the Green channel and inverse, both Blending> Multiply and New Channel. This gives you an Alpha channel, as back has none. I have done it for image _02
Scene file:
https://www.amazon.com/clouddrive/share/tMoFE9MFdeXlqvVcu5QC6KCIy9mPnTyX0gNRsg1jTpg
Screen shot
https://www.amazon.com/clouddrive/share/3kvU0Q10NWbVaO0rmc4jlO4otZ69EPbeeYQg2TS2hV5