### 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`
)
            latest_stream_arn = dynamodb_create_table(
                table_name=ddb_table, partition_key="id", stream_view_type="NEW_IMAGE"
            )["TableDescription"]["LatestStreamArn"]
            rs = lambda_client.create_event_source_mapping(
                FunctionName=function_name,
                EventSourceArn=latest_stream_arn,
                StartingPosition="TRIM_HORIZON",
                MaximumBatchingWindowInSeconds=1,
            )
            uuid = rs["UUID"]
            _await_event_source_mapping_enabled(lambda_client, uuid)
            assert poll_condition(is_stream_enabled, timeout=30)
            table = dynamodb_resource.Table(ddb_table)
            table.put_item(Item=items[0])
            
            retry(
                check_expected_lambda_log_events_length,
                retries=10,
                sleep=3,
                function_name=function_name,
                expected_length=1,
                logs_client=logs_client,
            )
            
            lambda_client.update_event_source_mapping(UUID=uuid, Enabled=False)
            time.sleep(2)
            table.put_item(Item=items[1])
            
            check_expected_lambda_log_events_length(