### 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`
event_source_uuid = lambda_client.create_event_source_mapping(
                EventSourceArn=stream_arn,
                FunctionName=function_name,
                StartingPosition="LATEST",
                BatchSize=num_records_per_batch,
            )["UUID"]
            _await_event_source_mapping_enabled(lambda_client, event_source_uuid)
            kinesis_client.put_records(
                Records=[
                    {"Data": json.dumps({"record_id": i}), "PartitionKey": "test"}
                    for i in range(0, num_records_per_batch)
                ],
                StreamName=stream_name,
            )
            events = _get_lambda_invocation_events(
                logs_client, function_name, expected_num_events=1
            )
            assert len(events) == 1
            lambda_client.update_event_source_mapping(UUID=event_source_uuid, Enabled=False)
            time.sleep(2)
            kinesis_client.put_records(
                Records=[
                    {"Data": json.dumps({"record_id": i}), "PartitionKey": "test"}
                    for i in range(0, num_records_per_batch)
                ],
                StreamName=stream_name,
            )
            time.sleep(7)  
            
            events = _get_lambda_invocation_events(