I totally fail to see what this has to do with LWJGL (or general Java game development)...
But to answer your question, this is not possible for the same reason that you cannot have Crysis run in your Java Swing JPanel.
Those are two different operating system processes using completely separate window system components.
While being two separate processes they also do not share the same virtual memory space and cannot communicate in any way with each other, unless specifically designed to do so.
The only way to do what you want is to have your Java application link against the JAR files of the Web Start application and invoke it programmatically. But for that to succeed, the Web Start application would very very likely need to be redesigned for such a usecase, in order for it to provide some sort of API for including its own window components into another application.