### 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`
result = testutil.connect_api_gateway_to_http_with_lambda_proxy(
            "test_gateway2",
            target_uri,
            path=lambda_resource,
            stage_name=TEST_STAGE_NAME,
        )
        api_id = result["id"]
        url = path_based_url(api_id=api_id, stage_name=TEST_STAGE_NAME, path=lambda_path)
        result = safe_requests.post(
            url, data=b"{}", headers={"User-Agent": "python-requests/testing"}
        )
        content = json.loads(result.content)
        assert lambda_path == content["path"]
        assert lambda_resource == content["resource"]
        assert lambda_request_context_path == content["requestContext"]["path"]
        assert lambda_request_context_resource_path == content["requestContext"]["resourcePath"]
    def test_malformed_response_apigw_invocation(self, create_lambda_function, lambda_client):
        lambda_name = f"test_lambda_{short_uid()}"
        lambda_resource = "/api/v1/{proxy+}"
        lambda_path = "/api/v1/hello/world"
        create_lambda_function(
            func_name=lambda_name,
            zip_file=testutil.create_zip_file(TEST_LAMBDA_NODEJS_APIGW_502, get_content=True),
            runtime=LAMBDA_RUNTIME_NODEJS14X,
            handler="apigw_502.handler",
        )
        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"
        result = testutil.connect_api_gateway_to_http_with_lambda_proxy(
            "test_gateway",