### 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`
assert len(event["event"]["Records"]) == num_records_per_batch
                actual_record_ids = []
                for record in event["event"]["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
                    assert "awsRegion" in record
                    assert "kinesis" in record
                    record_data = base64.b64decode(record["kinesis"]["data"]).decode("utf-8")
                    actual_record_id = json.loads(record_data)["record_id"]
                    actual_record_ids.append(actual_record_id)
                actual_record_ids.sort()
                assert actual_record_ids == [i for i in range(num_records_per_batch)]
            assert (
                invocation_events[1]["executionStart"] - invocation_events[0]["executionStart"]
            ) > 5
        finally:
            lambda_client.delete_event_source_mapping(UUID=uuid)
    def test_kinesis_event_source_trim_horizon(
        self,
        lambda_client,
        kinesis_client,
        create_lambda_function,
        kinesis_create_stream,
        wait_for_stream_ready,
        logs_client,
        lambda_su_role,