SAM invoke won't take local env vars

I have a sample SAM application with basic endpoints. I just want to run it locally by:

sam local invoke -e events/event-post-item.json putItemFunction --profile myprofile -n local.json

local.json is as follows:

{
    "Parameters": {
        "TABLE_CUSTOMERS": "MyDynamoDBTable" // MyDynamoDBTable is the DynamoDB resource name from `template.yml`
    }
}

And following is the code for putItemFunction

const dynamodb = require('aws-sdk/clients/dynamodb');
const docClient = new dynamodb.DocumentClient(); 
const tableName = process.env.TABLE_CUSTOMERS; 

exports.putItemHandler = async (event) => {
    const { body, httpMethod, path } = event;
    if (httpMethod !== 'POST') {
        throw new Error(`postMethod only accepts POST method, you tried: ${httpMethod} method.`);
    } 
    console.log('received:', JSON.stringify(event));  
    const { id, name } = JSON.parse(body); 
    const params = {
        TableName: tableName,
        Item: { id, name },
    };
    await docClient.put(params).promise();

    const response = {
        statusCode: 200,
        body,
    };

    console.log(`response from: ${path} statusCode: ${response.statusCode} body: ${response.body}`);
    return response;
};

I run this, and I get a "resource not found" error. I have made sure that the profile details are correct. I do understand that I am invoking it locally, and my local machine has a runtime docker only. Not the DynamoDB table, 'cause that's created over the cloud. Shouldn't SAM figure out the generated database name as defined in the Stack's template.yml file? When I hardcode the generated database name in the env file, it works. But that's not pragmatic approach. I want SAM to just use the table name that was just generated when I deployed the stack.