My suspicion is that you are having synchronization problems between MathJax and GWT. I don't know GWT at all, so can't really help with that, and the code you posted in your previous thread had most of its code minified, so it is pretty difficult to make sense of it. Is it possible for you to generate an un-minified version?
One difference between your jsbin version and the GWT one is that the latter uses an iframe, and it appears that MathJax is running within the iframe. The MathML seems to be loaded into the iframe, MathJax run, and then the results moved to the main document. This is not really a use-case that MathJax supports, and there are several issues you have to be concerned about. For example, MathJax will add styles to the iframe that would need to be copied to the main document, and may not be.
More importantly, I suspect that GWT may be moving the contents of the iframe before MathJax has finished processing. That could account for the results you are getting with Firefox and IE: MathJax first sets up spans for all the equations on the page, and then goes back and fills them in (there are performance reasons for that). I suspect that GWT is copying the page in between those two phases, before MathJax has typeset the math. The timing of when this occurs can depend on the browser, which is why it works in some and not others (it has to do with when the various signals for DOM content loaded and page load events occur, and those differ from browser to browser).
The [Math Processing Error] that you are getting may be for a similar reason. For example, it could be that GWT is tearing down the DOM of the iframe while MathJax is still working on it, and so data that it needs is missing when it goes to create the final equation on the page, and so gets the processing error. (The "w" from the message is likely to be part of bounding box data that MathJax stores attached to the elements in the page, so if they are being removed or altered while MathJax is working, that could cause the problem.)
I suspect that you need to coordinate the actions of MathJax with those of GWT. For example, when you queue MathJax's Typeset action via
MathJax.Hub.Queue(["Typeset",MathJax.Hub,element]);
the typesetting is not complete when the Queue function completes. So the on click event that calls MathJax.Hub.Queue() returns before the typesetting is done. GWT may think that that means it is time to copy of iframe to the main page, even though MathJax isn't done yet. So you may need to do something more to tell GWT to wait longer before the page is ready. Since I don't know GWT, I can't tell you what that is, but if there is some other way to signal GWT when things are ready, you might need to queue a second function that does that.
As for trapping the error, throwing the error outright would cause MathJax to stop running and not do any more typesetting, which is why that isn't done. But there are several ways you can get at the error. For debugging purposes, the error object for the last [Math Processing Error] is stored in MathJax.Hub.lastError, so you can use MathJax.Hub.lastError.stack to get the call stack for the error.
Alternatively, you can register message hook for those errors and have a callback called when one occurs. E.g.,
<script type="text/x-mathjax-config">
MathJax.Hub.Register.MessageHook("Math Processing Error",function (message) {
console.log(message[2].stack); // message[2] is the error object
});
</script>
I don't think you will learn anything really from this, as I don't think it is a MathJax-specific issue, but a timing issue between MathJax and GWT. But you might be able to look at the iframe DOM (e.g., console.log(document.body.innerHTML)) when the error occurs and see what is happening.
I hope that gives you something to try.
Davide