AWS/SentinelHub - Crop before download

I am not sure I understand what you want. .SAFE product only makes sense when you use the complete scene.
If you would like to get just the part of the Sentinel-2 tile, check the Remote Pixel’s tiler for guidance:

GitHub - RemotePixel/remotepixel-tiler: Sentinel / Landsat / CBERS serverless...

Sentinel / Landsat / CBERS serverless tile server. Contribute to RemotePixel/remotepixel-tiler development by creating an account on GitHub.

You can obviously get AOI-clipped data using Sentinel Hub, but I am not sure this is what you are after:

How can I clip the image to a specific polygon?

How can I clip the image to a specific polygon? You can add a parameter GEOMETRY as in given example. Simplify geometry to avoid exceeding maximum number of characters in URL.