OK, I nailed the solution for this rather convoluted problem.

In case someone else faces the same issue in the future I have used a combination of javascript Window.postMessage() and .Net ClientScriptManager.RegisterClientScriptBlock(Type, String, String, Boolean) to achieve DNN UI update.

Logic behind:

My remote app was modified with RegisterClientScriptBlock() to register execution of window.opener.postMessage() after DNN had communicated successful processing of the earlier message indicating that DNN UI needs to update. window.opener is the key here.

Inside my DNN module all I added was logic required to update UI itself inside of receiveMessage(event) function and relevant event listener.

Should you fear post messages from untrustworthy sources be sure to use

if (event.origin !== "http://example.org")
    return;

And thats all it took in the end.

PS: At first I tried to use javascript window.onunload event to trigger postMessage() but that had the effect of updating UI much too soon, in fact as soon as I triggered the POST request not giving adequate time to actually process the request. Otherwise the UI was updating just fine when I closed the window or triggered a subsequent request.