### 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`
return make_error_response("Unable to find path %s" % invocation_context.path, 404)
    
    validator = RequestValidator(invocation_context, aws_stack.connect_to_service("apigateway"))
    if not validator.is_request_valid():
        return make_error_response("Invalid request body", 400)
    api_key_required = resource.get("resourceMethods", {}).get(method, {}).get("apiKeyRequired")
    if not is_api_key_valid(api_key_required, headers, invocation_context.stage):
        return make_error_response("Access denied - invalid API key", 403)
    resource_methods = resource.get("resourceMethods", {})
    resource_method = resource_methods.get(method, {})
    if not resource_method:
        
        
        resource_method = resource_methods.get("ANY", {}) or resource_methods.get(
            "X-AMAZON-APIGATEWAY-ANY-METHOD", {}
        )
    method_integration = resource_method.get("methodIntegration")
    if not method_integration:
        if method == "OPTIONS" and "Origin" in headers:
            
            return get_cors_response(headers)
        return make_error_response(
            "Unable to find integration for: %s %s (%s)" % (method, invocation_path, raw_path),
            404,
        )
    
    invocation_context.resource_path = extracted_path
    invocation_context.integration = method_integration
    return invoke_rest_api_integration(invocation_context)
def invoke_rest_api_integration(invocation_context: ApiInvocationContext):