It's hard to say where this could be going wrong. Is there a exception stacktrace that you can paste and maybe that will give enough hints?
The API vendor and I tracked this issue down to what is described here:
They are considering an update to their API that remedies this, but who knows how long that will take.
Is there anything I can configure in CXF that would help fix this?
I received an updated third party API that included the necessary CXF configurations, and a jboss-deployment-structure.xml to include in my deployment.
I don't have time to dig through what they did, so I'm marking this reply as an answer to close out the thread.