Can I add an argument in the Process API request to add custom band names or band names from the source as-is (.e.g, VV and VH) that will appear in the FID or Attribute in the screenshot?
This isn’t something you can currently do with Process API.
I would recommend using GDAL and Python to rename the bands for you. A quick search found this post on the GIS Stack Exchange which should help you achieve your aim.