I was unaware of the details of your setup and only assumed you had already created the invoice items and that was your custom state list value.
In terms of what you are trying to do, which is essentially creating a shopping cart, although it is called an Invoice, it is the same as an Order for a shopping cart with the invoice items basically being order line items. You can not do this with custom states well.
I just put together a shopping cart without touching database using the plugin below.
Please be aware, I have nothing to do with this plugin. I learned about it about 5 months ago and only today had the use case for it and it is great for this type of stuff.