Actually, I think I figured it out - by default, cpanel's vhost template looks something like this for a site with a dedicated ip address:
Litespeed would not answer requests for the site in question that way. Once I changed the directive to look like this:
It seems to work fine. The only issue is that while the site works ok as long as it's called using the URL, if you call it in a web browser using just the IP address you still get a default cpanel page rather than the site bound to that IP. Anyone know of a fix for that so the site can be called using it's URL or the IP address directly?