Once I learned this is a known bug with ESRI, we came up with a klugy work-around.
Since we only needed to print secure services within our building, our solution was to:
1) Add specific DENY rules to our external firewall for the services we needed to print.
2) Using the siteadmin app, unsecure the services we needed to print.
3) Now printing works fine.
The firewall rules make it so only users inside the building can get to the services. Since that is true, we can unsecure them so that printing will work from inside the building. That meets our requirements.
ken.