From what I understand, you would like your Users on website to login via either Cognito or Shopify Credentials. The Shopify Login button on the other hand should be displayed on the Cognito Login page.
We have an application on the Shopify app store, called Store as IDP, which is the quickest way to integrate Shopify (as an Identity Provider) and AWS Cognito. Once the integration is completed successfully, whenever your users try to login on your website, they can use either their Cognito Credentials or Shopify Store Credentials.
Our application supports all the standard authentication protocols, such OAuth 2.0, SAML 2.0, JWT, OpenID, API Authentication, etc. and can be easily configured and integrated. Using our application, you can achieve the solution to your use-case.