If CloudWatch has a REST API, then you can use ServiceCallout in PostClientFlow to send the message to the endpoint. The endpoint needs to be publicly accessible, publicly routable.
I think Apigee documentation states that the only policy permitted for use within PostClientFlow is the MessageLogging policy. BUT, Apigee operations can also configure your organization to allow ServiceCallout in PostClient Flow. To make this change, contact your Apigee support rep. There's no charge for that.
So if you can figure out how to make a POST call to your CloudWatch endpoint, with the appropriate credentials and payload, then ... you should be able to use ServiceCallout to emit that call for every request that Apigee handles.
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