If a CNAME isn't feasible at this point, you could try using hosts file entries to acheive the desired result. The key thing to keep in mind is that the cookie has to be written in a domain that the community application can read. So if you're running your login page locally and don't already have a hostname setup for your local machine, you will likely need a hosts file entry for that as well.
Also, if you're setting up a cname or hosts file entry for a Lithium demo site, there is some configuration that our technical operations team will need to do in order to ensure the requests get routed properly. I'd suggest speaking with your sales representative regarding getting that setup. Or if you have access to our case portal, you could file a case with our support team.