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On iOS I need to sign in to a third party account using Sign in With Apple, but do not want to use the Apple ID I use on that device for iCloud. Instead I would like to use the Apple ID I use there for purchases. How do I select the Apple ID to use for signing in Sign in With Apple?

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I don't think you can. You have to use your iCloud login

From https://support.apple.com/en-gb/HT210318

You need to be signed in with your Apple ID in System Preferences > Apple ID on your Mac, or Settings > [your name] on your iPhone, iPad or iPod touch. You also need two-factor authentication turned on. Learn more.

and that is your iCloud login (I have the same setup as you a iCloud login and another for buying apps and Music so I can see which one is in Preferences.) The learn More link there goes to your link https://support.apple.com/en-gb/HT211687 which includes

To use Sign in with Apple, you need to use two-factor authentication and be signed in to iCloud with that Apple ID on your Apple device.

and

You need to be signed in with your Apple ID in System Preferences > Apple ID on your Mac or Settings > [your name] on your iPhone, iPad or iPod touch. You also need two-factor authentication turned on.

So this is a service of iCloud and not Apple Id

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