Completing checkouts using the Storefront API

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I've been looking into the Storefront API with the goal being to create a checkout, manage items and then complete the purchase. I've successfully been able to prototype a solution with all of this functionality except for the final stage, completing the payment. From reading the documentation (here and here) it looks like this stage of the checkout API is only accessible to public apps registered as a sales channel - is there any other way to achieve this? Can access be granted for a private app?