You can't invoke your function directly with an API Key. You need to implement a proxy layer that check your API Key and perform a request with OAuth2 granted identity token. To achieve this, you can use Cloud Endpoint or its brand fresh serverless implementation API Gateway. I wrote an article on Cloud Endpoint and you can reuse it on API Gateway.
If it's just for Postman and your tests, you can generate a token with the GCLOUD CLI
gcloud auth print-identity-token
Copy the result and add it to the header of your request
Authorization: Bearer <token>
It is valid for 1H. Perform your tests, when it is expired, generate a new one and continue.
I also wrote a small tool for this. Perform a precall with Postman to get the token and then use it in your request as previously described
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