I notice that you did not give a real example, and I'm not going to cover all the hypothetical possibilities; there are a variety of techniques for a variety of real-world websites.
The safe, cross-browser, works almost every time it's tried, method is Script Injection:
function GM_main () {
    var newVar = WEBSITE.uniqueObject.prototype.uniquePrototype;
    /* Or, explicitely...
    var newVar = window.WEBSITE.uniqueObject.prototype.uniquePrototype;
    */
    // DO WHATEVER WITH newVar HERE.
}
addJS_Node (null, null, GM_main);
//-- This is a standard-ish utility function
function addJS_Node (text, s_URL, funcToRun, runOnLoad) {
    var D                                   = document;
    var scriptNode                          = D.createElement ('script');
    if (runOnLoad) {
        scriptNode.addEventListener ("load", runOnLoad, false);
    }
    scriptNode.type                         = "text/javascript";
    if (text)       scriptNode.textContent  = text;
    if (s_URL)      scriptNode.src          = s_URL;
    if (funcToRun)  scriptNode.textContent  = '(' + funcToRun.toString() + ')()';
    var targ = D.getElementsByTagName ('head')[0] || D.body || D.documentElement;
    targ.appendChild (scriptNode);
}
Two variations of this were also linked from your previous question.
However, the most common cases of people wanting to use unsafeWindow seems to be to defeat ad-display timers or to trigger javascript that is normally triggered by a link or button.
In the common button/link scenario, don't break the sandbox with unsafeWindow.  Just programmatically click or mousedown the control.
In the case of cheating a website timer, since it is one line (not counting the hack), this is one case where unsafeWindow might be a good fit.  EG:
unsafeWindow.payTheBillsTimerCounter = 0;
Beware:
- Angry webmasters can theoretically exploit unsafeWindow.
- Please don't violate any Terms of Service (TOS).
- Please support websites whose resources you use (a lot).
- For Chrome userscripts and content-scripts, the unsafeWindowhack will probably be blocked circa Chrome version 28.  For straight scripting on Chrome, switch to Tampermonkey.  Tampermonkey is likely to keep supportingunsafeWindow, without any hacks needed (on your part).
 Tampermonkey also provides near-perfect Greasemonkey compatibility and a host of features that Chrome userscripts do not.