What happens when you try the {JFBCLike} tag? You mention it doesn't parse correctly, but we'll need more details on what is, or isn't, happening. I looked at the page you linked to and could see any tags or the underlying HTML for the Like button. If it's on a sub-page, let us know a specific one we can see the tag on so we can investigate further.
As for the module, I understand you said one won't work for you. I'm assuming that's because you don't have a module position in the right spot, but either way, I wanted to make sure you knew about the JFBCSocialShare module which can add the sharing buttons. It can also be used to create the 'floating' bar of buttons like on the bottom page of this site.
Thanks,
Alex