Agreed, nothing wrong with it. It's just redundant, as a swordsman, by definition, cuts things with his sword. However, since it's a bad idea to go around hacking up trees with one's sword (as it's not made for that task) it might not be clear from swordsman what tool was used.
So it's semantically odd -- suppose he used an axe? Then why call him a "swordsman" at all?
Anyway, the real awkwardness comes from the passive voice. If you change it to the active voice, you can write the sentence to include an explanation:
Since no other tool was available, the swordsman used his sword to chop down a few branches to make a fire, silently cursing the hours he's have to spend sharpening it afterward.