I don't know anything about Basque etymology, but the Latin word
asciola looks like a diminutive form from
ascia (adze, carpenter's axe, hatchet). I don't know how frequently the diminutive form was used--it does have the virtue of making the Basque word look possibly related to the Latin word since "l" and "r" are related sounds.
Maybe asciola and aizkora are evidence that the Seminole Indians sailed across to Spain in ancient times and introduced the people to the hatchet, which was then named in honor of the Seminole war leader Osceola.