AWS VPN Client with fixed EIP for interfaces

I see that you have a problem connecting to your AWS Client VPN service because the IP address associated with it is continuously changing,

This is an expected behavior of the AWS VPN Client service, and as such, it is recommended that the customer uses the DNS provided by Amazon as stated in the Client VPN documentation under the limitations and rules section of the documentation found here: https://docs.aws.amazon.com/vpn/latest/clientvpn-admin/what-is.html

Additionally, if you want to route out the traffic from the client network interface, you can create a NAT gateway inside the same VPC as the client interface, but in a public subnet. Similar scenarios and their recommended architecture can be found here: https://aws.amazon.com/blogs/networking-and-content-delivery/using-aws-client-vpn-to-scale-your-work-from-home-capacity/