At the moment, authors are at a terrible disadvantage relative to publishers with regards to this issue, since there's no collective, organized, resistance action to what publishers demand (that I know of, anyway). So you are not likely - from my limited experience, anyway - to get far with these negotiations; why should they make an exception for you? Lots of other fish in the sea.
I'd say the most important thing you can do is circumvent the copyrights issue: Publish a version of the paper which is exactly the version you initially submit to the journal on your website, university paper repository, or ArXiv, possibly with an appropriate license (e.g. see the ArXiv license page) - but not with the template used for the journal submission, i.e. as a plain-vanilla article.
Having done so, surrendering your rights via a contract witht he publisher no longer means all that much, because everybody in the world has the rights you allowed them with your free license - including yourself. The paper will thus be publicly-accessible and updatable by you without the journal being able to say anything about it. The only thing you will likely not be able to do is re-publish it in another for-pay journal etc.