How can I use the dotnet ef cli within AWS .NET Lambda functions with container images

Question

I am following these steps https://docs.aws.amazon.com/lambda/latest/dg/csharp-image.html#csharp-image-clients to create a custom image that I want to run in a Lambda Container. I created this docker image

# You can also pull these images from DockerHub amazon/aws-lambda-dotnet:8
FROM mcr.microsoft.com/dotnet/sdk:8.0

# Set the image's internal work directory
WORKDIR /var/task
RUN dotnet tool install --global dotnet-ef 
ENV PATH="${PATH}:/root/.dotnet/tools"
# Copy function code to Lambda-defined environment variable
COPY "bin/Release/net8.0/linux-x64"  .
  
# Set the entrypoint to the bootstrap
ENTRYPOINT ["/usr/bin/dotnet", "exec", "/var/task/bootstrap.dll"]

As you can see, I want to use dotnet ef commands in my container in AWS Lambda, therefore I install it globally. Then, I have the following function handler in which I want to run dotnet ef commands

public class Function
{
    /// <summary>
    /// The main entry point for the custom runtime.
    /// </summary>
    /// <param name="args">Command line arguments.</param>
    private static async Task Main(string[] args)
    {
        var handler = FunctionHandler;
        await LambdaBootstrapBuilder.Create(handler, new DefaultLambdaJsonSerializer())
            .Build()
            .RunAsync();
    }

    /// <summary>
    /// A simple function that takes a string and does a ToUpper
    ///
    /// To use this handler to respond to an AWS event, reference the appropriate package from 
    /// https://github.com/aws/aws-lambda-dotnet#events
    /// and change the string input parameter to the desired event type.
    /// </summary>
    /// <param name="input">The input to the Lambda function handler.</param>
    /// <param name="context">The ILambdaContext that provides methods for logging and describing the Lambda environment.</param>
    /// <returns></returns>
    public static async Task<string> FunctionHandler(string input, ILambdaContext context)
    {
        // How To use dotnet ef command here?
        return input;
    }

}

but somehow dotnet ef is not installed. Is that even possible?