You can solve the cold start problem of Java Lambda by compiling your code to native with graalvm native-image. In that case you will need the Custom Runtime.

I invite you to check another post I wrote on the subject https://frank-afriat.medium.com/lets-compile-your-aws-lambda-in-java-eafc24aae269

You may also check the framework I created for this purpose : https://microlam.io