We bundle the Java 6 JRE with our application installer so that it can be run on any machine, but this makes the application a little bit heavier. So we are planning to red
You're trying to reduce a standard JRE's size? Don't do that. You can choose to bundle an alternative JRE which might be smaller. A list can be found on this Wikipedia page. As always, beware of compatibility issues and test your application thoroughly.
An other, and safer, way is to just require an installation of a JRE on the target machine.