You can re-think the design of having the banks in the region communicating to your servers via IP.
If the banks communicate using DNS names, you have much more flexibility to move your servers around.
You can also achieve improvements in high availability and resiliency by moving to DNS connections.
So a plan might be
- Setup a DNS record for your existing server
- Get the banks who connect to your server to connect via the DNS name
- Setup your new server in the other account (other answers describe this)
- Cut the banks over to your new server in the new account simply by updating the DNS record
I haven't tried load balancing across accounts, but that may be another option, which would give you HA as a bonus. By registering your current instance, and new instance in another account as targets with a load balancer and getting your clients to connect to the load balancer, you could cut over to the other account. The only part I haven't tried is registering targets in different accounts, but looks like this should be possible with an AWS Network Load Balancer