### 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`
res = sqs_client.receive_message(QueueUrl=destination_queue)
                msg = res["Messages"][0]
                body = json.loads(msg["Body"])
                assert body["requestContext"]["condition"] == "RetryAttemptsExhausted"
                assert body["DDBStreamBatchInfo"]["batchSize"] == 1
                assert body["DDBStreamBatchInfo"]["streamArn"] in stream_arn
            retry(verify_failure_received, retries=5, sleep=5, sleep_before=5)
        finally:
            lambda_client.delete_event_source_mapping(UUID=event_source_mapping_uuid)
class TestLambdaHttpInvocation:
    def test_http_invocation_with_apigw_proxy(self, create_lambda_function):
        lambda_name = f"test_lambda_{short_uid()}"
        lambda_resource = "/api/v1/{proxy+}"
        lambda_path = "/api/v1/hello/world"
        lambda_request_context_path = "/" + TEST_STAGE_NAME + lambda_path
        lambda_request_context_resource_path = lambda_resource
        create_lambda_function(
            func_name=lambda_name,
            handler_file=TEST_LAMBDA_PYTHON,
            libs=TEST_LAMBDA_LIBS,
        )
        
        lambda_uri = aws_stack.lambda_function_arn(lambda_name)
        target_uri = f"arn:aws:apigateway:{aws_stack.get_region()}:lambda:path/2015-03-31/functions/{lambda_uri}/invocations"