The airport will not be as busy as during the rush hour at night so getting to the airport at least 2 hours before departure should be fine. Not much to do in the departures area, a few shops and a couple of fast food outlets (Subway). The security should be fine - make sure to get a tag for your carry on bags so it can get stamped at the security check point. You'll also have to go through immigration so have your forms filled out.

I think Druk Air operates an A319 on this route via KTM. Try to get a seat on the left side (Seats A-C) for great views of the Himalayas if the weather permits.
Have a great time in Bhutan.