You can do what you want to do with the normal Calibre install. The only prerequisite is that you can access the library folder from both devices, if you've set it up as a shared folder on your network, that should be fine. Point both Calibre installations to the network folder as their library folders. You said you've already set up your main computer like this, you'll need to point the laptop at the shared folder, then, and tell it that there's already a library present in that folder.
There's a big caveat attached to this method: You'll need to make absolutely sure that both Calibre installations aren't running at the same time. If you try to access the library with one computer while the other is already running Calibre, you'll be in a world of hurt.
Another solution would be to use the content server and
EPUBReader (a Firefox plugin) or similar.