AWS WebSocket API: 'Internal Server Error' with Increased Connected Clients

From your comments it looks like you have both throttling and errors in your Lambda function. We have a great many resources describing throttling and how to overcome it:

By default, there are 1000 units of concurrency allocated to each AWS account per region. This capacity is shared by all the functions in your account, per region. What this implies is that no function in your account can be invoked simultaneously by a rate of more than 1000 times. And if more functions are being invoked at the same time, they will share from the 1000 available units

https://docs.aws.amazon.com/lambda/latest/operatorguide/throttling.html

https://aws.amazon.com/blogs/compute/understanding-aws-lambdas-invoke-throttle-limits/

https://docs.aws.amazon.com/lambda/latest/dg/invocation-retries.html

Hello,

Concurrency is a crucial factor to consider in AWS Lambda, as it can lead to application failures through a phenomenon known as Throttling. By default, each AWS account is allocated 1000 units of concurrency per region, shared among all functions in that region. This means that no function can be invoked more than 1000 times simultaneously, and if more invocations occur simultaneously, they will share from the available 1000 units. However, this limit can be increased by requesting support from AWS. Throttling in AWS Lambda happens when function invocations surpass the available concurrency units, resulting in the function not running and triggering a RateExceeded exception.

The following links should help you understand and give you more insights on how to troubleshoot throttling issues

https://repost.aws/knowledge-center/lambda-troubleshoot-throttling

https://aws.amazon.com/blogs/compute/understanding-aws-lambdas-invoke-throttle-limits/