We could transfer files with FileZilla using FTP over TLS and that's what we were trying to replicate. We have found a solution.
Our cURL issue turned out to be two fold. (1) We added cURL option --ftp-ssl. However we then ran into a certificate issue. (2) The certificate we had (and they need) was not sufficient. We were missing some intermediate level certificate. I'm not sure what specific certificate, as we added quite a few in one shot while troubleshooting.
Thank you for your help!