I would like to use only Cognito User Pool, and therefore I want to use identity federation with Cognito User Pools, without Cognito Federated Identities (identity pools).>
Based on your comment to Summer Guo, here's what it seems like you're having an issue with...
A lot of details missing here, but if you're using a Cognito Authorizer in your API Gateway, then it doesn't know about any 3rd party IdP -- it just knows about your Cognito User Pool. So sending the CUP JWT will work, while sending anything else won't. If you want to use a Google auth token, then you need to implement a Custom Authorizer that verifies this token with Google.
I presented on this topic during reInvent. Here's the video that goes into the details: https://www.youtube.com/watch?v=VZqG7HjT2AQ