Which is better against Ganon: Master Sword + Barbarian set or an Ancient weapon + Ancient set?

Ignoring the no pause playthrough

Master Sword has 60 base damage against Ganon. With the Barbarian set, that increases by 50% to 90 damage.

The Ancient weapon most similar to the Master Sword is the Ancient Short Sword (both are one-handed swords). It has 40 base damage, which automatically increases by 50% to 60 against Ganon. So without clothing sets, the two weapons have equal base damage against Ganon. However, the Ancient Proficieny bonus from the Ancient set is 80%, increasing damage to 108, larger than that of the Barbarian set.

So an Ancient Short Sword + Ancient set deals superior damage to Ganon than a Master Sword + Barbarian set. The Master Sword does have the advantage of not breaking and can be obtained without rupees. However, the Ancient proficiency also extends to other Ancient weapons such as the Ancient Bladesaw and Ancient Bows and can be used alongside the Ancient Short Sword. In terms of damage output, Ancient weapon + Ancient set is the clear winner.

With the no pause playthrough, it is much less straight-forward

The only Ancient items that can be used are the Ancient arrows (not technically a weapon, the bow is) and the Ancient Short Sword by the Weapon Connoisseur mission. So we can only compare the Ancient Short Sword against the Master Sword, but with the Ancient proficiency bonus applied to both the Short Sword and Ancient Arrows.

The base durability of the Ancient Short Sword is 54. This effectively means that it can deal 5832 damage to Calamity Ganon before breaking, assuming that all hits are used on Ganon, use up one durability each, and no durability is used elsewhere. Calamity Ganon has 8000 HP, and loses 1000 HP for each Divine Beast (min of 4000).

The Master Sword has much more durability (200) and can last the whole fight. However, with the Ancient Short Sword, you can at least pick up the weapons in the Calamity Ganon area, whereas this is not possible with the Master Sword if you waste durability and it becomes inert during the fight. Of note, you can deal damage to Ganon in other ways such as Arrows (this is where the Ancient Arrows helps) and Urbosa's Fury. The Master Sword does have the advantage that you can get it, waste the durability to make it inert, and then use it again for other things, giving it more use than just the Ganon fight. However, the Ancient Short Sword practically mandates you to go to Ganon and use it solely for that fight if you want to use it against Ganon.

Considering all this, if the goal is to go to Ganon immediately after getting the weapon, and you have cleared at least two Divine Beasts, then I feel Ancient Short Sword + Ancient Set is better. At 6000 HP, Ancient Sword comes extremely cause to finishing Calamity Ganon by itself, but I feel this is still in its favour since the remaining <200 HP can be cleared with Ancient Arrow, Urbosa's Fury, and the nearby weapons. If you have cleared 0 or 1 Divine Beasts, it depends how many Ancient Arrows you intend on using, which I consider to be more subjective.