Sorry but what does "There is no offline LAN option" exactly mean? I extensively play most of my games in LAN, with son, Bro in law and nephews............
For Elemental you will have to have you, your son, bro in law, nephews etc all create an Impulse Account. On that impulse account they have to purchase a copy of the game. You can install using whatever copy, but for multiplay each player must have their own independent copy of Elemental registered on an Impulse Account.
The "No offline LAN option" means that when you begin a LAN game with your family, the game is being run by a Stardock Server remotely. Stated plans from Stardock are to eventually add the ability for you to be your own server (ie you could set up your game just for your family, without having to connect to stardock), based on demand.
In other words, right now you are not going to be able to play Elemental as a family without buying multiple copies of the game, and waiting for Stardock to finish setting up their servers (which should be within the week).