What is the related Apache directive used. Apache may have a mod_resin, which is not available in LiteSpeed.
For Litespeed, I think should use HTTP reverse proxy to talk to backend Resin.
There probably is not any easy solution here. You need to read mod_caucho and resin documentation see if it is possible to have resin start a HTTP listener on one port, then let LiteSpeed talk to resin via HTTP proxy.