The settings in your screenshot are meant to be changed by store staff, and so currently it's not possible to change this setting using the API.
If your app requires this setting to be enabled, you can query the shop object and look for the
taxes_included field to determine whether or not this setting is enabled for a shop. If it's not enabled, you can disable app functionality and/or display a message to the user advising them to enable the setting.
JB | Developer Support @ Shopify
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