Believe you can have your business flow with just 1 SO, multiple Delivery orders and multiple MOs

Each order (project as you refer) can have the products (Complete order) with a MTO route. On confirmation of SO a Delivery order and MO for ‘all’ of the order qty would be created.

You can produce part quality in MO and create a backorder, also delivery the part qty and create a Backorder
The SO would keep track of delivered qty and each time a delivery is done you can raise an invoice (multiple invoices linked to 1 SO)

You can identify the SOs which requires to be invoiced from the field Invoice Status as it would change to ‘To Invoice’. This field is visible on the sale orders list view.

Not clear how the project App integrated in your business flow.