Tomcat doesn't sound like a good choice for me in your situation. AFAIK it's primarily made for Servlets and JSPs, and you have neither. You also don't need to deploy multiple applications on your app. server etc. (so no benefit from ".war").
If you need dependency injection, connection pooling, logging, network programming framework etc., there are a lot of good solutions out there and they don't need tomcat.
For example, in my case I went for a standalone app. that used Spring, Hibernate, Netty, Apache Commons DBCP, Log4j etc. These can be easily setup, and this way you have a lot more freedom.
Should you need a HTTP server, maybe embedding Jetty is another option. With this option too, you have more control over the app. and this can potentially simplify your implementation compared to using a tomcat container.