Is it possible to set AlwaysVisibleControl invisible?

Yep. The control still renders out as whatever the control renders as (if that makes sense). e.g. you could apply an AlwaysVisibleControlExtender to a div with runat='server' or a panel with runat='server' and in either case they'd render in the html as a div with the same id (unless using master pages in which case you'll have to do a little more work to get the unique id).

So, since I like declarative markup, I'd setup a div or panel with an id (let's say id='alwaysVis'), and then I'd grab that control in the xml-script and set a couple properties thusly:

<control id="alwaysVis" visibilityMode="collapse" visible="true" />

Then I'd have a button insdie that div or panel, let's name it id='closeButton' that I'd grab in the xml-script thusly:

<button id="closeButton"> <click> <setProperty target="alwaysVis" property="visible" value="false" /> </click></button>

Hope that helps.Paul