Lambda function inside VPC cannot use client.post_to_connection | It works outside the VPC

OVERVIEW OF THE SYSTEM: I want to use the following configuration, however, I an open tu suggestions:

Client <-> API Gateway (Websocket) <-> Lambda <-> RDS DB (Postgre)

The idea is to create an API which I can call from my React application, which will access my database.

• The application needs to read and write from the database (via the API).
• The application needs to be scalable
• The system needs to be secure
• This is a learning project, but I am training to make it as close to a real application as possible

THE ISSUE: If the Lambda function is outside the VPC, it cannot access the database and I get a timeout error. If the Lambda function is inside the VPC, it can access the database but it cannot communicate repsonses back to the client - it throws this error:

[ERROR] ForbiddenException: An error occurred (ForbiddenException) when calling the PostToConnection operation: Forbidden
Traceback (most recent call last):
  File "/var/task/lambda_function.py", line 48, in lambda_handler
    response = client.post_to_connection(ConnectionId=connectionId, Data='response')
  File "/var/runtime/botocore/client.py", line 530, in _api_call
    return self._make_api_call(operation_name, kwargs)
  File "/var/runtime/botocore/client.py", line 960, in _make_api_call
    raise error_class(parsed_response, operation_name)

SECURITY DETAILS:

• My security group is set to allow all outbound traffic, but it is restricted to allow inbound traffic only on certain ports and from some IP addresses.
• I did set up an internet gateway

import psycopg2
import json
import os
import json
import urllib3
import boto3
from datetime import datetime
from psycopg2.extras import RealDictCursor

client = boto3.client('apigatewaymanagementapi',endpoint_url=os.environ['END_POINT'])

class DateTimeEncoder(json.JSONEncoder):
    def default(self, obj):
        if isinstance(obj, datetime):
            return obj.isoformat()
        return super().default(obj)

def lambda_handler(event,context):
    body = json.loads(event["body"])

    if 'categories' not in body:
        body['categories'] = ['gender','publicPolicy']

    connection = psycopg2.connect(
    host=os.environ['HOST'],
    database=os.environ['DATABASE'],
    user=body["credentials"]["username"],
    password=body["credentials"]["password"]
    )
    print('Connected to database')
    categoriesString = ", ".join(["'{}'" for _ in body["categories"]]).format(*body["categories"])
    connectionId = event["requestContext"]["connectionId"]

    try:
        with connection.cursor(cursor_factory=RealDictCursor) as cursor:
            cursor.execute("SELECT * FROM questions WHERE category IN ("+categoriesString+");")
            results = cursor.fetchall()
    except Exception as e:
        print("Error occurred:", str(e))
    finally:
        connection.close()

    responseMessage = json.dumps(results,cls=DateTimeEncoder).encode('utf-8')
    print(responseMessage)
    print(connectionId,os.environ['END_POINT'])
    response = client.post_to_connection(ConnectionId=connectionId, Data='response')

    return {"statusCode":200}