Use single AWS Cognito User Pool with multiple "Sign in with Apple" integrations?

Question

I currently have three different apps, and I want the users from all three apps to be able to sign in to the same Cognito User Pool using "Sign In With Apple".

I've set up the "Services ID Configuration" on Apple Developer for one of the apps, and then entered the resulting Identifier, etc., under the Federation settings in Cognito. Using this configuration works, and all three apps are able to sign in as expected.

However; this means all three apps get presented with the same app logo and name during the sign up process. I don't see a way around this, as there is seemingly no way to specify more than one configuration under "Federation"/"Sign in with Apple".

Am I missing something? Or will I have to create additional User Pools?