OK time for a definitive answer from a specialist. I hope you'll accept it as I'm a mere university Science Dean at a Russell Group university right now and hold Biochemistry Chairs in the UK and USA. It's quite a few years now since I was a Chemistry Reader at a university down the road but I'll give it my best shot. Hope it's good enough!
The K2SO4 solution is acidified with HCL for two reasons:
The first is that BaSO4 tends to crystallise in very tiny particles that have a tendency to occlude the formation of impurities. The acidification of the solution allows larger crystals of BaSO4 to form unhindered.
The second reason is that you have to remember that barium ions will precipitate other anions such as carbonates present in the K2SO4 as impurities. Acidifying the solution will largely prevent the precipitation of any carbonates, sulphites (as Teddio mentions) etc present.