API Gateway set stage variable based on source IP Address

API Gateway does not support content based routing. One option might be to invoke a Lambda function that will invoke the appropriate backend.

A potential design could be that you create an API Gateway backed by a Step Function. (https://docs.aws.amazon.com/step-functions/latest/dg/tutorial-api-gateway.html).

Then you could pass the IP or get it as part of header/payload (encrypted if you wish), passit to a Choice step that based on that decides which Step/Lambda function to call.