Hi Mr. Izak,
To me, it seems like you're describing a self-signed certificate. In this case, I'd second Denis's comment and further state that you're going to get that warning from Chrome and you won't see a security lock on your browser. This is because you are telling the network that you are you but no other device on the network can confirm that. Thus, it's not strongly secure.
I know each shop is different and I'm not sure if you have an Apache or IIS web server residing in front of the Tomcat server itself, but my suggestion would be to get the network folks to create a cert, based off of the URL that you're utilizing, and then add that into your web or application server (IE: Apache Web Server or Tomcat Application Server). From there, you should see the security lock on your browser. This is because all of the relevant network devices (IE: DNS & proxy servers) will have that information and can validate the URL. It's what I'm used to and what I currently utilize (Minus the steps to walk through utilizing OpenSSL for Apache).
As far as the concern if it's there or not, that depends on the particular shop. If they're security minded (IE: Public sector), it will be a concern for them. If they're not concerned with SSL communication, it won't be a concern. I would suggest determining if it is a requirement or not and moving forward from there.
Let me know if this was what you're looking for. Thanks,
-Victor