### Is there an existing issue for this?

- [X] I have searched the existing issues

### Current Behavior

I Created an REST APIGatway with lambda handler using AWS CDK. when I hit the API the response header content type is always  set to "text/html"  instead of `application/json` . even if I changed the response template or added the header to lambda returned response. when I deploy the same stack on AWS account  the response header is fine and work as expected

### Expected Behavior

Response Header content-type to be `application/json`

### How are you starting LocalStack?

  `localstack start` command

### Steps To Reproduce

#### How are you starting localstack (e.g., `bin/localstack` command`)

    `localstack start` command

#### Client commands (e.g., AWS SDK code snippet, or sequence of "awslocal" commands)

I'm using AWS CDK typescript application and deploy it using `cdklocal` command on localstack

Using CDK follow the steps:
- create  REST ApiGateway
- create Lambda function 
- integrate lambda with APIGatway

On Command line:
- deploy using `cdklocal deploy` command

Finally
- test the endpoint and check the response header content-type




### Environment

```markdown
- OS: Windows 
- LocalStack: latest
```


### Anything else?

When I deploy the same stack on AWS account everything works fine and the response header content-type is `application/json`
expected_length=1, function_name=function_name, logs_client=logs_client
            )
        finally:
            lambda_client.delete_event_source_mapping(UUID=uuid)
    
    def test_deletion_event_source_mapping_with_dynamodb(
        self, create_lambda_function, lambda_client, dynamodb_client, lambda_su_role
    ):
        try:
            function_name = f"lambda_func-{short_uid()}"
            ddb_table = f"ddb_table-{short_uid()}"
            create_lambda_function(
                func_name=function_name,
                handler_file=TEST_LAMBDA_PYTHON_ECHO,
                runtime=LAMBDA_RUNTIME_PYTHON36,
                role=lambda_su_role,
            )
            latest_stream_arn = aws_stack.create_dynamodb_table(
                table_name=ddb_table,
                partition_key="id",
                client=dynamodb_client,
                stream_view_type="NEW_IMAGE",
            )["TableDescription"]["LatestStreamArn"]
            result = lambda_client.create_event_source_mapping(
                FunctionName=function_name,
                EventSourceArn=latest_stream_arn,
                StartingPosition="TRIM_HORIZON",
            )
            event_source_mapping_uuid = result["UUID"]
            _await_dynamodb_table_active(dynamodb_client, ddb_table)