That is a tall order: a reliable, first-hand account of a one-on-one fight where the armed gladiator wins!

I'm not sure if this checks off all of the requirements, but the best I could think of was Martial's On the Spectacles. This is about the opening games of the Flavian Ampitheatre, better known as the Colosseum. The poem is most famous for the Priscus v Verus fight, but Martial also discusses beast-fighting.

That which was the highest glory of thy renown, Meleager, how small a part is it of Carpophorus' - fame, a stricken boar! He plunged his hunter's spear also in a headlong-rushing bear, the king of beasts beneath the cope of Arctic skies; and he laid low a lion, magnificent, of bulk unknown before, one worthy of Hercules' might; and with a far-dealt wound stretched in death a rushing pard. He won the prize of honour; yet unbroken still was his strength.

You can look through it here .