Lightsail - Private hosted zone - subdomain Route53

Just to be clear, your private zone would be for internal DNS resolution to your VPC, this would not be a public application, correct?

The goal is to create subdomain.domain.com and host a private application in lightsail VPS.

We already have a public application hosted in lightsail VPS at domain.com

Ok, I haven't tried this so I'm not sure if it will work, but you would need to use Route53 for private hosted zones and have VPC peering enabled with the Route53 associated VPC. I believe it will give you a permissions error if you try to add the Lightsail VPC to the route53 association.