The problem here is which ever the application first starts, loads the dll file and second application sees this. The easiest fix (if possible) is to run the the two applications in different JVMs. In websphere you can do this by creating a different Application Servers though Websphere Admin Console (browser based). Each Application Server runs in a different JVM and won't cause this issue.
But if you need to run both apps in the same server you may need to investigate whether you can share a dll loaded by another class loader. Try catching the UnsatisfiedLinkError and NOT throwing it in both apps so that the second application will try to work without loading the dll. (note that UnsatisfiedLinkError is an Error not an Exception, So catch UnsatisfiedLinkError or Throwable instead of catching Exception).