I would give it to the bastard sword.
Even though its shorter than the longsword. the leverage incurred by it is significantly greater than the katana. The bastard sword is not only a slicer and stabber. it can smash pretty effectivy to. Something the katana cannot do. The power behind each stroke will rattle the katana user if the blades clash continuolly. I think the katana wielder will win based upon the swordfighting style used as superior.But the performance of blade will go to the bastard sword.
It strong enough not to break. short and light enough to prevent fatigue. But concentrates alot of power.
However. just something for you to ponder about.
The rapier and epee are both extremly weak blades and hold nothing to the japanese katanas. Yet they are the most effective dueling swords ever created. There are accounts of duels bettween katanas and and fencers and the fencer almost always comes out on top. I like the scenario you pick. it seems pretty fair. But we have to understand the katana is overated...people make it out to be the ultimate sword. I believe the duel is totally dependent on the style and the sword working together. The epitome of that is fencing.
I would argue that the rapier is the weakest sword but probably the most deadly.
But hey...im a fanboy of the katana...its pretty badass. But when it comes down to it the weapon is an allaround weapon. Some might see that as a strength. But it has no particular strengh itself, which leaves it prey to any weapon that has somethng really strong goign for it. With longswords its power strokes. With the rapier is the ever so deadly thrust.
But the katana is only good all around. Guess thats what makes it such a popular choice.
I guess me being right or wrong is dependent on one thing, Is an all around sword better. or a specialized sword.
i hope this helps and expands your thoughts on the matter
please rate and thank you.