What you want to achieve isn’t possible in Shopify or any rate-providing app when using multiple Profiles. What you see when a customer checks out with items from both locations is a “subcart”. Shopify will split the cart and give each the rate that the Profile has designated. They will then either mesh the two rates together and display 1 rate, or display both rates and the customer will pick between (usually the cheapest) that will apply to both Profiles.
Shopify will only ever allow 1 checkout process to display unlike other online retailers.
Your only option would be to offer delivery when items in the cart are “delivery” when “pickup” items are with it too, however, this would also require a rate-providing app to do and all items inside 1 general profile.
I hope that helps answer your question. If you want or need further information, please do ask me.
Adding to what Aaron said: the feature is now documented in the Shopify Help Center under “Split shipping in Shopify Checkout.” Worth noting it is currently for Plus and Enterprise plans, so if you are on a standard plan it may not be available yet.