At this stage the easiest way to add this functionality is if a third-party app is installed on your store. An app such as the Connected Inventory app here is one that can help synchronise any inventory changes to variants or products that are within a connected group. Just a note however that apps such as this one are made by third-party developers outside of Shopify. So if you have any questions about them I would recommend getting in touch with the app developers, as they are best equipped to answer app-specific queries.
I definitely see how being able to sync inventories across bundled products or variants is a useful option to have within our admin, so I will send this as a feature request to our developers. As we are constantly looking to improve our platform, perhaps this is something we can incorporate in the future. Please note that although I will be submitting this request, we are unable to track it as this is an internal process that our developer team oversees. However there is currently a public facing changelog here that you can check for new features that has been added to Shopify.
Was there a specific reason that you're not wanting to integrate a third-party app onto your store? Third-party apps are often the solution for stores looking to add a specific functionality that isn't yet available natively within the Shopify admin. We have a guide here on third-party apps if you'd like to learn more about it. Alternatively however, if you still would like a solution without a third-party app involved, you can look into hiring one of our Shopify Experts to help you code a tailored solution onto your store's theme.
You would set up your variations to be similar to this:
Option 1: Pillow only
Option 2: Pillow and insert
The problem here is Shopify doesn't have a native kit/bundle option for products so you would need to use an app to automate the process of calculating your inventory correctly.
The other option you have is to customise your theme to create custom product pages whereby the user can select separate products to add to a bundle. There are apps available for this method but I believe you could also modify your theme to make this work without an app.