Pointing AWS EC2 elastic IP to instance in VPC

It seems like you may be assuming this is more complicated than it is.

You should be able to access anything on the Internet in general -- including and especially EC2 instances -- from within a VPC by using a NAT instance, which is likely how your VPC machines are already accessing AWS services like S3, SQS, SNS, etc., or external resources like downloading software updates.

http://docs.aws.amazon.com/AmazonVPC/latest/UserGuide/VPC_NAT_Instance.html

You don't need a tunnel, unless you need the traffic to be encrypted, which seems unlikely (since you could just use SSL) ... and if you do have a tunnel, you'll still need a NAT instance in order for your GRE traffic to make it out to the EC2 machine and the responses to come back... unless your Nginx machine already has an external public IP address of its own... in which case, you don't need anything, because you already have the fastest possible connectivity between the VPC machine and the EC2 machine.

This is probably because the database caches the URL name.

This could be resolved easily via few SQL commands. Changes are required in three of your tables:

wp_options, wp_posts and wp_postmeta

UPDATE wp_options SET option_value = replace(option_value, 'http://www.oldurl', 'http://www.newurl') WHERE option_name = 'home' OR option_name = 'siteurl';

UPDATE wp_posts SET guid = replace(guid, 'http://www.oldurl','http://www.newurl');

UPDATE wp_posts SET post_content = replace(post_content, 'http://www.oldurl', 'http://www.newurl');

UPDATE wp_postmeta SET meta_value = replace(meta_value,'http://www.oldurl','http://www.newurl');

Ref: https://wpbeaches.com/updating-wordpress-mysql-database-after-moving-to-a-new-url/

Hope this helps.

Peace.