I suggest you keep that ProductID field instead of an externally generated Part number. That doesn't keep you from doing what you want here really and the relationship is just a bit better to work with in the long run.
Add the Part Number field to the Products table. You can add a field for it to line items if you want, but this isn't absolutely necessary. You can either add the part number field from the products table to the Line Items portal and also to the invoice print layout that's based on Line Items or you can define a field in the line items table set to look up (copy) the part number from products. In which case you would add that field to your layouts.
Your drop down field's value list can be modified to include the part number so that when you select a product, you can see and select by part number, but the value list still enters the product ID in order to link the line item record to the correct record in products.