### 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`
return (
                dynamodbstreams_client.describe_stream(StreamArn=latest_stream_arn)[
                    "StreamDescription"
                ]["StreamStatus"]
                == "ENABLED"
            )
        function_name = f"lambda_func-{short_uid()}"
        ddb_table = f"ddb_table-{short_uid()}"
        items = [
            {"id": short_uid(), "data": "data1"},
            {"id": short_uid(), "data": "data2"},
        ]
        try:
            create_lambda_function(
                func_name=function_name,
                handler_file=TEST_LAMBDA_PYTHON_ECHO,
                runtime=LAMBDA_RUNTIME_PYTHON36,
                role=lambda_su_role,