Heres a checklist to help get started:

1) download the php 5.3.X sources and build --with-litespeed -> build ls-php binary
2) Study any available litespeed lsapi protocol documentation

3) Study the nginx fast-cgi module, how it forwards the requests
4) Study the 'how to write and nginx module' guide for writing an nginx module
5) Modify the existing nginx fast-cgi module (and replace the fcgi-specific parts with lsapi interface)

It may help to study these lsapi requests and responses directly by attaching to the TCP port.
i.e. Tracing the conversations between webserver and the ls-php program (ethereal)

You may also try to debug / step the code in php-src/sapi/litespeed.
This is OSS code available under BSD license.