I get it, but I have several identical objects in the scene. The amount of scenarios would be pre-established.
Would have a way of following the player to create the level, I return to the menu without losing one layout?
If all your puzzle objects are identical, then it makes it even easier. The same concept applies. You want to get your data into an Array, and then save it as JSON text. In your case, you can use WebStorage and set a local value where the key = "Level [X]", and the value = [JSON data]. In the end, you'll have three local values saved, one for each level. When you want to load a given level: simply fetch the data using the respective key, load from JSON into an array, and then loop through the array to fetch the position values, and set your objects' positions accordingly.
Note that if you're not used to using Arrays, it might be worth getting some extra help. If you don't like this approach, I would explore the current built-in Save functionality that C2 provides (as @procrastinator mentioned), but I don't know much about it, and I believe you mentioned that the built-in Save does not work for you.