Lambda container images complain about entrypoint needing handler name as the first argument

It depends what the language is of the runtime. For example, if it is NodeJS, then the handler name should look like:

"app.handler"

If it is Java, then it should look like:

"com.example.LambdaHandler::handleRequest"

The image will look for them in LAMBDA_TASK_ROOT so you will need to make sure that your code (or compiled code) is copied to that folder when you build the image, for example:

COPY target/* ${LAMBDA_TASK_ROOT}