### 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`
kinesis_create_stream(StreamName=stream_name, ShardCount=1)
            wait_for_stream_ready(stream_name=stream_name)
            stream_summary = kinesis_client.describe_stream_summary(StreamName=stream_name)
            assert stream_summary["StreamDescriptionSummary"]["OpenShardCount"] == 1
            stream_arn = kinesis_client.describe_stream(StreamName=stream_name)[
                "StreamDescription"
            ]["StreamARN"]
            uuid = lambda_client.create_event_source_mapping(
                EventSourceArn=stream_arn, FunctionName=function_name, StartingPosition="LATEST"
            )["UUID"]
            _await_event_source_mapping_enabled(lambda_client, uuid)
            kinesis_client.put_records(
                Records=[
                    {"Data": record_data, "PartitionKey": f"test_{i}"}
                    for i in range(0, num_events_kinesis)
                ],
                StreamName=stream_name,
            )
            events = _get_lambda_invocation_events(
                logs_client, function_name, expected_num_events=1
            )
            records = events[0]["Records"]
            assert len(records) == num_events_kinesis
            for record in records:
                assert "eventID" in record
                assert "eventSourceARN" in record
                assert "eventSource" in record
                assert "eventVersion" in record
                assert "eventName" in record
                assert "invokeIdentityArn" in record