not sure if you can do a more test to confirm it's dedicated IP issue.
SSL cert can bundle with a domain name in litespeed, not necessary always with an IP.
can you make a configuration change in httpd.conf (I don't know yet since no test env), so that not assign a dedicated IP to https virtual host, see if this site is ok after cold start. (you can try this on command line: /usr/local/lsws/bin/lswsctrl stop; wait a few seconds; ensure "ps -ef|grep litespeed" is empty; then "/usr/local/lsws/bin/lswsctrl start", so without rebooting the server)
or a simple way, to check /usr/local/apache/logs/error_log, if any log message in it after litespeed start up.