If you set-up product P as a BoM type Kit. You would be able to buy Product P while receive its components.

On the receipt operation the component serial numbers can be updated. With this you would also be able to view Product P and the components in inventory.

But, you wouldn't be able to provide sl number for Product P. To do that you would need to manufacture product P with a BoM type manufacture. All 4 serial would be available from the traceability report. 

Once product P is manufactured on hand qty of product P would increment, while components would no longer be on hand. You would either have on hand qty of product p or the components not both. (It would amount to double counting the component)

It is possible to have more than 1 bom for a product, do ensure to have KiT BoM is the 1st BoM for product P.

Additionally, while selling product p, delivery order would be for components and would need to manually add product P (manufactured BoM) while not delivering the component.

While in theory the above solution is possible, I would NOT recommend it. You might want to have product P as a KiT or as a finished (manufactured) product and not have it as both.

From the post it is not clear how product P is handled through the various stages, is it dealt with as product P or as bundle of components? Is it sold or further used as a component in other product BoM.

Depending on the specific need that requires addressing it is possible to do so in V17