While trying to debug the above 409 response (which is an undocumented response) I discovered this:
https://developer.apple.com/documentation/appstoreconnectapi/add_a_scheduled_price_change_to_an_app
On this page it describes a way to add a price change to an app as a scheduled event. The documentation is incomplete however because it does not show a sample of the POST payload. Instead it links to the 'object' that is to be sent but that documentation is a rabbit hole of objects that are documented very poorly as well. An example payload would be a much better way to document this call.
However, on that page there is a large red box with a warning that says, "If you use this endpoint to add a scheduled price change to your app, you can’t use AppPriceInlineCreate to change your app’s price."
So YES there is a way to change your app's price in the API.
You do so by using the poorly documented AppPriceInlineCreate object.
But the question remains, what is the endpoint for making this request, and what exactly is Value: appPrices which is a required part of AppPriceInlineCreate