There has been discussion about ways to support this: Autocompletion Service Support for external text editors, although I'm not aware of anyone working on it currently.
Any solution to this is going to need to be tied directly into the engine. No offense, but your suggestion to use the .rst files is a bad one. First of all, the API docs are all autogenerated anyway, so it would be an unnecessary step - you should pull directly from the source. Secondly, any changes to the .rst formatting would break your tool, so it would be a huge pain to manage.
See here for how the documentation is generated/organized: Contributing to the documentation
Personally, I think integrating the Language Server Protocol or something similar is really going to be the best solution. It will be flexible and support a variety of editors. If you are able to contribute to that, I think it would be the best use of your time.