How to call Greasemonkey’s GM_ functions from code that must run in the target page scope?

To use Greasemonkey’s GM_ functions from code that must run in the page scope (Such as your timeBtn click handler), do the following:

1. Have the page-scope code use postMessage to send the data in string format.
2. Have the Greasemonkey script listen for the appropriate messages and call the desired GM_ function(s) with the message data.
3. Use JSON to safely package data in strings.

Adding window.postMessage () and window.addEventListener ("message"... to your code, it becomes:

... ...

//-- Only run in the top page, not the various iframes.
if (window.top === window.self) {
    var timeBtn         = document.createElement ('a');
    timeBtn.id          = "gmTimeBtn";
    timeBtn.textContent = "Time";
    //-- Button is styled using CSS, in GM_addStyle, below.

    document.body.appendChild (timeBtn);

    addJS_Node (null, null, activateTimeButton);

    window.addEventListener ("message", receiveTimeMessage, false);
}

function activateTimeButton () {
    var timeBtn = document.getElementById ("gmTimeBtn");
    if (timeBtn) {
        timeBtn.addEventListener ('click',
            function () {
                var ytplayer = document.getElementById ("movie_player");
                /*-- GM_functions will not work here, so send the data
                    back to the GM script scope.
                */
                //-- Tag the message, we may not be the only ones sending.
                var messageTxt  = JSON.stringify (
                    {currentVidTime: ytplayer.getCurrentTime ()}
                );
                window.postMessage (messageTxt, "*");
            },
            false
        );
    }
    else {
        alert ("Time button not found!");
    }
}

function receiveTimeMessage (event) {
    var messageJSON;
    try {
        messageJSON     = JSON.parse (event.data);
    }
    catch (zError) {
        // Do nothing
    }

    if ( ! messageJSON  ||  ! messageJSON.currentVidTime)
        return; //-- Message is not for us.

    /*--- We have a time value, set it with GM_setValue ()
        But, WARNING: First make sure that the stored value is
        a safe string.  GM_setValue() crashes on just about anything else.
    */
    var safeValue       = JSON.stringify (messageJSON.currentVidTime);
    GM_setValue ("videoMarkedTime", safeValue);
    console.log ("Video time recorded with GM_setValue ().");
}

... ...

You can see the stored value by opening about:config and searching for videoMarkedTime.