What reaction will ensue when sulfur trioxide is added to excess barium hydroxide solution?
Basically yes.
Sulfur trioxide reacts with water to form sulfuric acid (this is extremely dangerous!  do not try it at home) in a strongly exothermic reaction which, unless done under very precise control, produces a fine mist of concentrated sulfuric acid vapor (not nice).
$$\ce{SO3 + H2O -&gt; H2SO4}$$
If you are doing this with an excess of barium hydroxide then the sulfuric acid will be deprotonated in two steps as you suggest.
$$\ce{H2SO4 + OH- &lt;=&gt; HSO4- + H2O}$$ $$\ce{HSO4- + OH- &lt;=&gt; SO4^2- + H2O}$$
Barium sulfate will precipitate from the solution, driving the reaction to completion.
$$\ce{Ba^2+ (aq) + SO4^2- (aq) -&gt; BaSO4 (s)}$$
I couldn't find any mention of barium hydrogen sulfate as a compound anywhere so I take this to mean that it is not stable and therefore will not precipitate.