Maybe I don't understand the question. To be more clear, if you shutdown Tomcat (and hence it's JVM), then neither Tomcat nor it's connection pools, nor any of it's daemon threads would still be alive ... and thus no memory references would be left in tack.
How exactly have you come to the conclusion that a Tomcat resource has a reference to a connection pool, after Tomcat is shutdown? You wouldn't be able to see that in a profiler, and frankly, it doesn't quite make sense. For example, is it possible that the OS keeps connections open for a short period of time before destroying them? Have you tried the same tests using the same OS, but instead with a different container, something like Jetty? Are you using keep-alives, or some form of persistent connections?
If you want to get down-and-dirty with Tomcat's management of a resource exposed through JNDI, you can make your own DataSource implementation that delegates to the H2 DataSource. You can do this my using the factory-method attribute within your server.xml (or context.xml)
Here's a working example on github : https://github.com/tvollmer/connection-factory
Beyond that, it would help if you could further clarify what you mean by "various issues on shutdown". Those details matter, and it's not clear to me how you've logically gone from "various issues" to saying that Tomcat "does not dispose the connection pool". You could be dealing with 2 completely separate issues here, and you may want to check out antiResourceLocking and antiJARLocking settings for Tomcat on Windows.