I have found an easier method. Export the data, re-install Jasper using the .exe installer and the bundled Tomcat and then simply overwrite the Tomcat files with the new version.
I am using Jasper Reports Server 6.2.1 which comes bundled with Tomcat 8.0.14. This version of Tomcat has a known vulnerability to Path Traversal Attacks. I want to upgrade just the version of Tomcat. How do I do this?