When I was looking for a utility to export the non-exportable private keys in Windows, I found the mimikatz tool, which enabled me to do that and a lot more.
To export the private keys, run mimikatz as administrator and type:
crypto::capi crypto::certificates /export
And you’ll get the certicates exported with the password mimikatz. You can also export the machine certificates with
/systemstore flag. See the wiki for more info.
This tool is detected as a threat by many antivirus, so you’ll have to probably disable yours before using it.