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).
$$\ce{SO3 + H2O -> H2SO4}$$
If you are doing this with an excess of barium hydroxide then the sulfuric acid will be deprotonated in two steps as you suggest.
$$\ce{H2SO4 + OH- <=> HSO4- + H2O}$$ $$\ce{HSO4- + OH- <=> SO4^2- + H2O}$$
Barium sulfate will precipitate from the solution, driving the reaction to completion.
$$\ce{Ba^2+ (aq) + SO4^2- (aq) -> BaSO4 (s)}$$
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.