I don't think you'll want to package the H2 files within your WAR application. It's not going to be able to use them in the WAR.
What you want in your version control is the DDL used to build the database entities (and pre populate if necessary). Then, either in your development or production environments, you use these scripts to create/update the database.
What you have in your web application is simply the JDBC url (specifying the file location of the database) used to locate the embedded database. As you suggested, this will be different based on development or production environments.
I've been using LiquiBase as a way of managing database creation/upgrades, and it's pretty easy to use.
However, the idea of creating the database as part of development and then embedding those files within the WAR in order to use them directly won't work. There's no concept of saving those files, so none of the data would persist.