That's a pretty interesting question. This isn't actually an answer, but this is where I would start.

1) Google for "servlet proxy" to find an existing implementation of a proxy servlet, either from Apache/Tomcat or a 3rd party.
2) Install and configure the servlet in your LabKey webapp and test (ignoring security)
3) If that works, modify/wrap the servlet to handle any LabKey specific requirements (e.g. security and/or container mapping)

This might be a good example for the community.

Matt