What do these iATS - SOAP errors mean?
Turns out the issue was our server's operating system. We were on CentOS 5.5, and it doesn't support OpenSSL higher than 0.9.8. We had to upgrade to a new server. We decided on a CentOS 7 server and are in the process of migrating. The CentOS 7 server came with OpenSSL 1.0.2 installed. I setup and tested our connection with iATS today and had it working in no time at all. Hooray!

While I had manually updated OpenSSL, I neglected to update it with our server's package manager (and our website would only recognize what the package manager was using). But it turns out with CentOS 5.5, the yum package manager only goes up to 0.9.8e. There was no way around it.

There are so many benefits to CentOS 7 over 5.5, even with all the migration work, I'm glad we made the change.