How to upgrade AWS API Gateway endpoints without API interruptions

Question

I have been tasked with upgrading the servers / endpoints that our AWS APIGateway uses to respond to requests. We are using a VPC link for the integration request. These servers are hosted on AWS Elastic Beanstalk.

We only use two resources / methods in our API : /middleware and middleware-dev-4 that go through this VPC link.

As our customers rely heavily on our API I can not easily swap out the servers. I can create new servers and point the APIs to those but I do not want any downtime in our API service. Can you recommend a way to do this API change without impacting our customers ?

Iv'e seen several examples using canary release but they seem to pertain to Lambda functions and not VPC links with EC2 servers as endpoints.

Edit --

AWS responded and they agreed that I should add the new servers to the target group in the network load balancer and deregister the old ones.