Headless Shopify Plus stores have a store and checkout that are separate. This means that when a customer goes from cart to checkout, the store URL usually changes.
With a headless setup, the cart and the checkout are separated and run a different URL. This prevents the app from adding a product to the cart when ‘add to cart’ is clicked.
You can fix this by upgrading to the new Shopify Checkout Extensibility. You’ll be able to add your upsells using the Checkout Editor in your admin settings. The add to cart is native to this checkout.
Want help setting this up? Reach out in the chat here or in the app!