What kind of ancient warfare? Disorganized Germanics with clubs and axes and naked Celtic Barbariens? Or more organized Phalanx or Maniple Soldiers of the Greek and Roman?

For the first, the Barbarians sadly left little documentation. Here I would actually trust the ARMA / HEMA experiments on this. Sadly, they are late medieval/Renaissance focused and not "ancient". First hand tests I heard about seam to suggest, that if you never fought a Southpaw, this is indeed an advantage for your opponent. ... Which is why actual medieval manuals do touch the topic of fighting left-handed opponents: http://www.hroarr.com/manuals/sabre-cutlass-broadsword-rapier/PallasArmata.pdf (page 54 onward). With sufficient training, the left hand offered no advantage.

There is another form of ancient combat that focuses more on man-against-man fighting then formations and unlike the barbarians, this one is somehow documented: Gladiators. Now, it's a stretch to say, that Gladiator combat represented warfare of the time but I think it's as close as you can get. According to this exhibition of the West Cheshire Museum: https://pbs.twimg.com/media/CpXdfAJW8AAs2_3.jpg:large there is one depiction of a left-handed Gladiator and that is the only one. This can have many reasons: The mason of this relief might have made a mistake, the masons of other reliefs made mistakes, or actually, there were only very few gladiators fighting left-handed (certainly less then the 10% of the population, you'd expect, if it made no difference). However, I guess it's quite certain to say, that left-handedness was no big think, that could be considered advantageous for a Gladiator, or this would have been more prominent.

I think the most likely reason is found in the fact, that many Roman Gladiators represented Roman or Greek Soldiers, which brings me to the other form of ancient combat: Formations.

And here, left-handedness was certainly a disadvantage because in a Phalanx or Maniple, you had to carry your shield in your left and the sword or spear in the right, no exceptions. That just does not work for an orderly formations, thus naturally left-handed men were at a disadvantage because they were forced to fight with their weaker hand.

(I can't really support the lack of exceptions since I don't know every source, so instead, here's some pictures from Trajan's Column without even one left-handed soldier: http://www.webphoto.ro/poze/columna/roman_soldiers.jpg - http://www.webphoto.ro/poze/columna/battle.jpg - http://www.webphoto.ro/poze/columna/columna_traian.jpg )