AWS Client VPN to improve cross region performance
I don't think this will work. A Client VPN lets you into your Sydney VPC, but it's not transitive to your us-east-1 resources. AWS Global Accelerator might help, and CloudFront would probably help but you've said you don't want a CDN.
You'd probably be better off working out why your website isn't working properly in Australia. https://www.webpagetest.org/ lets you test your server from many points around the world, which might help, but you should also check your web server error logs, security group rules, and anything else that might block it.