Will it work with the open_basedir restrictions? That can't be disabled.
As for file uploads, I'm not sure why it's saying that. File uploads are enabled.
Do you know if Exponent CMS connects to the Exponent server and uploads files from there? That could be the issue. If you believe it is so, I can request that Exponent's server is whitelisted so that these uploads will work. If you are unsure, I would ask on the Exponent forum why this error is happening so I have more information to help troubleshoot your problem.