Hi Timo,
Clemency from Zapiet here, our Store Pickup + Delivery app could work for you 
On our Pro plan, you can create up to 200 locations that can act as individual pickup locations that the customer can choose from in the cart page. You can see how this looks from the customer side on our demo store.
You can then create one location in Shopify that will be your physical warehouse where your products are stocked. Our app does not take into account inventory so all pickup locations will show in the cart page whilst stock remains in your central warehouse. All orders will be fulfilled from that one Shopify location, no matter which pickup location they chose via our app.
As one location within Shopify would be fulfilling all orders a mixed cart would not be an issue. You can configure the locations within our app so that all products are available in each location. This just means that each location will show as available in the cart page no matter the products they have in the cart.
If you wish to have one flat free rate, you can configure this in your Shopify shipping and delivery settings. This will show in the checkout page after the customer has selected their pickup location via our app in the cart page.
I hope this is all clear to you. Please feel free to reach out to us if you have any further questions
Clemency Farmer
Merchant Support Specialist - Zapiet Ltd
support@zapiet.com / +44 20 4526 8806
Learn about new feature releases and updates at the Zapiet Blog