I am a little confused as to why you want the iFrame with the chat to cover the entire screen when the chat itself only occupies a small section at the bottom right of the page. Elements with a greater stack order is always placed in front of elements with a lower stack order... so to achieve what you are trying to do with an iFrame of size 100% 100% will more than lightly require a questionably lengthy workaround.
"Yes the following method does involve manipulating the exact size of the iframe"
I did something like this a few months ago and i simply placed the iFrame on the page using absolute-positioning. Since most of those 'Live Chat' windows allow scrolling for its content i just worked with two states/heights for the iFrame. The two heights of the iFrame must match the open and close heights of the chat window contained within the iFrame.
Now.. when the user clicks on the Chat Now button... the iFrame will call a function of its parent window that will manipulate the height of the iFrame. *Just be sure to add an ID tag to the iFrame to allow the OPEN and CLOSE functions (in the parent window) to be able to manipulate the height of the iFrame.
This worked well for me!
HTML
<div><button>Click Me</button></div>
<iframe id="ChatFrame" allowtransparency="true" frameborder="0" scrolling="no" src="https://www.botlibre.com/script?file&id=15069189" style="position:absolute; height:38px; width:165px; bottom:0px; right:0px; borde:0px; background:none;"></iframe>
JavaScript to be placed in the parent window
var LiveChatOpen = false;
function open_LiveChatWindow()
{
document.getElementById('ChatFrame').style.height=340+'px';
document.getElementById('ChatFrame').style.width=320+'px';
LiveChatOpen = true;
}//End of open_LiveChatWindow
function close_LiveChatWindow()
{
document.getElementById('ChatFrame').style.height=38+'px';
document.getElementById('ChatFrame').style.width=165+'px';
LiveChatOpen = false;
}//End of close_LiveChatWindow
Stuff to do on the page that is being placed via iFrame:
Now all you have to do is attach the onClick events to the buttons in the Chat-GUI that Maximizes and Minimizes the chat window.
Eg: Add an event listener to the "CHAT NOW" button that calls the open_LiveChatWindow() function of the parent window.
You can call the functions in the parent window like this: parent.open_LiveChatWindow();
I hope you found this solution useful.