If I understand well, you want to use EJB3/JPA within a servlet container.
There are not only stand-alone implementations of JPA, but also embeddable EJB3 container, such as OpenEJB or Glassfish embeddable container. So nothing prevents you from starting such an embeddable container from the Servlet container to use EJB3.
(Note: I don't know all the details about transactions. In a full-blown app. server, you have JTA and a distributed transaction manager. You don't have that in a Servlet container such as Tomcat. JPA works with JTA and plain JDBC, but I don't know exactly how an embeddable container work without JTA. Still, I guess that would work given that such embeddable containers are also meant for unit-testing where I guess there is no JTA distributed transaction manager.)
Another approach would be to use Spring. Spring and EJB3 have indeed become very similar. You can start the Spring DI container within the Servlet container and benefit more or less from the same facilities as EJB3 (declarative transactions, etc.). See this post about Spring vs. EJB3.
All these technologies have become pretty modular, especially with JEE profiles. You can use Sevlets, EJB3, JMS, JPA, even JTA somehow independently of one other. You can also create an environment where you cherry pick the ones you would like, either with Spring or with JEE. The question is when does it stop to make sense and rather use an app. server with everything available and easily manageable. I think Servlet/EJB3/JPA is the limit, if more is needed go for an app. server.