There should not be a problem implementing your solution.The RadToolTipManager's AJAX event should not fire in the case of pressing > or >> (unless these elements have been added to the manager's TargetControls collection).
Also, as long as the tooltip manager's TargetControls is kept in synch after each AJAX request that changes the state of the page, there should not be a problem.
It seems that you are trying to attach tooltips to certain dates in the calendar and fire AJAX events that fill the tooltip with extra information for the particular date. While this is not so trivial to do when AJAX is involved in the calendar, it should be possible to achieve.
Our suggestion is to turn off AJAX for the calendar and implement your scenario without it - then if you have any problems with the AJAX enabled, send us a fully runnable project and we will examine it and make the necessary changes for you.
All the best,
the Telerik team