I ended up resolving the issue on my own.
I downloaded the missing driver from https://customerconnect.vmware.com/en/downloads/details?downloadGroup=DT-ESXI70-INTEL-IGBN-1520&productId=974 and then I imported it into LCM. That cleared the download error.
I then disabled HA on the cluster and remediated one host at a time. After each remediation, the compliance check for the host failed with another error:
A failure occurred when starting a host compliance check operation on host 'host.domain.com' : Error: com.vmware.vapi.std.errors.internal_server_error Messages: com.vmware.esx.task.exec.error<Failed to start the task. Please ensure the system has enough resources and retry.>
What solved that for me was a 'services.sh restart'. Afterwards, I checked compliance against the cluster and it came back green. I was then able to enable HA on the cluster, and the agent was able to install on each host without any problems.
I still have a meeting with VMware Support on Monday to figure out why I can't export the image as JSON/ISO/ZIP. That, too, throws an error.