There's a way to force a specific installed version to run, but if one Java plugin replaces the other there might be an issue with this. See linked question for further details.
Force Internet Explorer to use a specific Java Runtime Environment install?
Are there any issues with running the 1.4 code on 1.6? If not, you should be fine. Just avoid using anything in the classpath that changed significantly between these versions. Otherwise, look [here][1]. The codebase download thing is interesting, since it lets you download a different classpath. So, regardless of the JRE version, you could force the download of a 1.4 JRE classpath, which should ensure full compatability.
Edit: found an updated guide to using previous classpaths with newer Java versions in applets, which actually seems to work:
http://download.oracle.com/javase/tutorial/deployment/applet/deployingApplet.html
Edit 2: I actually have it working, but it appears to need to download JRE 1.4 and install it, but then it will run the applet with it automatically. However, this might be suitable for your needs.
http://www.2shared.com/file/bl3Rua2e/applet.html -- extract the archive, and then run index.html inside this. All source code is included.