### 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`
try:
        response = invoke_rest_api_integration_backend(invocation_context)
        
        
        invocation_context.response = response
        response = apply_response_parameters(invocation_context)
        return response
    except Exception as e:
        msg = f"Error invoking integration for API Gateway ID '{invocation_context.api_id}': {e}"
        LOG.exception(msg)
        return make_error_response(msg, 400)
def invoke_rest_api_integration_backend(invocation_context: ApiInvocationContext):
    
    invocation_path = invocation_context.path_with_query_string
    method = invocation_context.method
    data = invocation_context.data
    headers = invocation_context.headers
    api_id = invocation_context.api_id
    stage = invocation_context.stage
    resource_path = invocation_context.resource_path
    integration = invocation_context.integration
    integration_response = integration.get("integrationResponses", {})
    response_templates = integration_response.get("200", {}).get("responseTemplates", {})
    
    relative_path, query_string_params = extract_query_string_params(path=invocation_path)
    integration_type_orig = integration.get("type") or integration.get("integrationType") or ""
    integration_type = integration_type_orig.upper()