How Can I Route Internet Traffic from a Private Instance in Region-B Through a Public Instance's Elastic IP in Region-A Over VPC Peering?

That isn't possible. VPC peering connections are strictly non-transitive, point-to-point connections between exactly two VPCs. Each VPC can only use its VPC peering connections to reach IP addresses that belong to one of the CIDRs of the peered VPC. IP addresses outside the peered VPC, such as IPs on the public internet, are not reachable over the peering connection.

To reach destinations on the public internet through a VPC peering connection, you'd have to use an indirect, non-routed method. In practice, you'd place a proxy server in VPC A and connect from VPC B to that proxy server. The proxy server's IP address would fall within VPC A's CIDR range(s), making it reachable from VPC B over the VPC peering connection. The proxy server would then connect to the internet via a NAT gateway or directly through an internet gateway or egress-only internet gateway, without passing through a VPC peering connection and without violating the non-transitivity rule of VPC peering connections.

Routing internet traffic from a private instance in one region through a public instance's Elastic IP in another region over VPC peering is not directly possible with the standard AWS networking setup. VPC peering is designed for private communication between VPCs and does not support transitive routing or internet gateway sharing across regions.

However, you can achieve a similar result by using a combination of AWS services and some additional configuration. Here's an approach you could consider:

1. Instead of VPC peering, use AWS Transit Gateway to connect your VPCs across regions. Transit Gateway supports inter-region peering and allows for more flexible routing configurations.

2. In VPC-A (Region-A), set up a NAT Gateway in the public subnet instead of using the EC2 instance (Instance-A2) for NAT. This provides a more scalable and managed solution for NAT.

3. Configure the route tables in VPC-B (Region-B) to send internet-bound traffic to the Transit Gateway.

4. Set up the Transit Gateway route tables to forward internet-bound traffic from VPC-B to VPC-A.

5. In VPC-A, configure the route table to send internet-bound traffic from the Transit Gateway to the NAT Gateway.

6. Ensure that the security groups and network ACLs allow the necessary traffic flow.

This setup would allow Instance-B1 in Region-B to access the internet through the NAT Gateway in Region-A, effectively using the same public IP address for outbound internet traffic.

Keep in mind that this configuration may introduce additional latency due to the inter-region routing and could incur higher costs due to data transfer charges between regions and the use of Transit Gateway. Also, you would need to ensure that your NAT Gateway can handle the combined traffic from both regions.

If you specifically need to use Instance-A2 as the NAT device (instead of a NAT Gateway), you would need to set up custom routing and NAT rules on that instance to handle traffic from both VPCs. This would be a more complex setup and may not be as reliable or scalable as using AWS-managed services like NAT Gateway and Transit Gateway.
Sources
VPC peering - Amazon Virtual Private Cloud Connectivity Options
Community | Multi-Region Networking on AWS