I think you're missing a core concept in Eclipse RCP, and that is OSGi. OSGi is a module system that requires each module (i.e., JAR) to import the dependencies it uses by package. Such entries are found in the JAR's MANIFEST.MF. While they can be manually edited to import these dependencies, usually a tool (such as bnd) makes this much easier to manage.
Eclipse PDE does much of this work for you (and so does adding the Spring project nature), but you need to add any dependencies as plugins (which are also OSGi modules) so that they're visible to the RCP product runner built within Eclipse. Simply adding them to your build classpath will not translate to them being included on a runtime classpath.
When you attempt to run your product / application within Eclipse, make sure you click on the "Validate plug-ins" button to ensure that they are all visible runtime to the target RCP app you're running.