Proxy certain http requests in 1 AWS EKS cluster via another AWS account

Question

I'm in the process of migrating a legacy application which runs on ECS on AWS account A to EKS in AWS account B. As part of this application, some pods need to periodically call out to a third party API via http/https. This API requires that the public IP of our traffic from the AWS account is whitelisted on their side. What we have found is that when moving the application into account B, these requests have started to fail since the Public IP addresses have changed (due to new Nat Gateways in Account B). Apart from whitelisting the new IP addresses from account B on the API side, I've been asked to look at a way to see if it's possible to forward the traffic which goes to this API endpoint via our old AWS account A, since those IP's are already whitelisted (This is a political business decision which is out of my hands much to my disagreement). I'm struggling to see how this may be possible and conceptualise this in my head. Would I need to deploy a forward proxy in the EKS cluster, would I need to peer the two AWS accounts together and set up routing between the two to handle this scenario are just some of the things i'm thinking about. Hope this makes sense but if anyone has any ideas on how/if this can be achieved, I would be grateful.