Auto Scaling AWS Lightsail
Solution 1:
As of 29th November 2017, you can now use load balancer with Lightsail.
Amazon Lightsail has added load balancers to its easy-to-use cloud platform, enabling developers to build scalable, highly available websites and applications easily and quickly. Load balancers can be launched in minutes, fully configured and ready to route traffic to Lightsail instances for a low, predictable price of $18 per month. Lightsail load balancers also allow customers to easily build and maintain secure applications that accept HTTPS traffic with free SSL/TLS certificates and intuitive, built-in certificate management.
See https://aws.amazon.com/about-aws/whats-new/2017/11/amazon-lightsail-adds-load-balancers-with-integrated-certificate-management/
Solution 2:
For this use case, you'll need to use the "real" EC2. Lightsail is designed for one-off servers.
Solution 3:
What you might want is Docker deployed to Amazon ECS on Fargate. This will autoscale as needed and you won't have to manage physical or virtual server resources.