How do I commit case-sensitive only filename changes in Git?

I took @CBarr answer and wrote a Python 3 Script to do it with a list of files:

#!/usr/bin/env python3
# -*- coding: UTF-8 -*-

import os
import shlex
import subprocess

def run_command(absolute_path, command_name):
    print( "Running", command_name, absolute_path )


    command = shlex.split( command_name )
    command_line_interface = subprocess.Popen( 
          command, stdout=subprocess.PIPE, cwd=absolute_path )

    output = command_line_interface.communicate()[0]
    print( output )

    if command_line_interface.returncode != 0:
        raise RuntimeError( "A process exited with the error '%s'..." % ( 

              command_line_interface.returncode ) )

def main():
    FILENAMES_MAPPING = \
    [
        (r"F:\\SublimeText\\Data", r"README.MD", r"README.md"),
        (r"F:\\SublimeText\\Data\\Packages\\Alignment", r"readme.md", r"README.md"),
        (r"F:\\SublimeText\\Data\\Packages\\AmxxEditor", r"README.MD", r"README.md"),
    ]


    for absolute_path, oldname, newname in FILENAMES_MAPPING:
        run_command( absolute_path, "git mv '%s' '%s1'" % ( oldname, newname ) )
        run_command( absolute_path, "git add '%s1'" % ( newname ) )
        run_command( absolute_path, 
             "git commit -m 'Normalized the \'%s\' with case-sensitive name'" % (
              newname ) )

        run_command( absolute_path, "git mv '%s1' '%s'" % ( newname, newname ) )
        run_command( absolute_path, "git add '%s'" % ( newname ) )
        run_command( absolute_path, "git commit --amend --no-edit" )


if __name__ == "__main__":
    main()