API Gateway blocking response from Lambda via CORS despite CORS headers in response.

When I attempt to POST to my REST API deployed to API Gateway, my Lambda does not return a proper response. Instead, it returns:

Access to fetch at 'https://dev.api.no-name-games.com/pay' from origin 'http://localhost:8080' has been blocked by CORS policy: No 'Access-Control-Allow-Origin' header is present on the requested resource. If an opaque response serves your needs, set the request's mode to 'no-cors' to fetch the resource with CORS disabled.

I read somewhere that if your API is a REST API with a Lambda integration, you can't set CORS via the console, so I set it in the Lambda itself:

return events.APIGatewayProxyResponse{
	StatusCode: 200,
	Headers: map[string]string{
		"Access-Control-Allow-Origin": "*",
	},
	Body: data.ClientSecret,
}, err

However, when I look at my request/response, I do not see those headers: