Set a Keyword for the new forge like "MyNewForge." Next, head over to the Constructible Object record category and create the item recipes, setting the Workbench Keyword setting to "MyNewForge." Then, when you use the forge, only your weapons will be displayed in the menu and not the defaults.
The organization of the crafting menu itself is hardcoded, but so long as you use the correct Keywords, you can set up your forge so that only your specific items show in the crafting menu for it.