The simple way to do is is to use the service
resource to stop Tomcat, use the bash
resource or a file resource to copy the WAR file, and then use the service
resource to start Tomcat again.
That cookbook you are trying to use doesn't do WAR file deployment. But I found this one - https://github.com/poise/application_java - that may do the job.
And for what it is worth, I think you could find a better Tomcat installation cookbook too. For a start, there is a "tomcat" recipe on the Opscode Community Cookbooks site that can handle Tomcat 7.
(Hint: don't just use the first cookbook you find with Google. Do a Github search, and try to assess the cookbooks functionality and quality)