The issue turned out to be certificate-related. The WCF service called by the console app uses an X509 cert for authentication, which is installed on the servers that this script is hosted and run from.
On other servers, where the same services are consumed, the certificates were configured as follows:
winhttpcertcfg.exe -g -c LOCAL_MACHINE\My -s "certificate-name" -a "NETWORK SERVICE"
As they ran within the context of IIS. However, when the script was being run as it would in production, it's under the context of the user themselves. So, the script needed to be modified to the following:
winhttpcertcfg.exe -g -c LOCAL_MACHINE\My -s "certificate-name" -a "USERS"
Once that change was made, all was well. Thanks to everyone who offered assistance.