I have looked into your issue this morning and I have come up with a work around it. I understand you want to use the Magento translated message in your system that consumes Magento API. I guess your concerns are not to duplicate the existing translation and instead enhance a unified system..
The translation don't seem to be built-in with the API layer in Magento; It may be reasonable to admit this is ok as loading translations has a cost and that would possibly incur api latency issues?
My solution was to add a new API that gives you the translations for the store you use; afterwards, you can cache the result of this translation API call. You can use this API in 2 ways:
retrieve a single translation:
http://<magento_url/index.php/rest/V1/gettranslation?message=The requested qty is not available&storeId=1
retrieve all the translations:
to be able to consume this additional API, you will have to use the module I have come up with, it is a public repository you can access at: