Two can play at that game. See 5:15-5:22.
Even if she didn't need to sleep or rest, Basch would doubtlessly be able to outrun her (a man of his physique against a female archer?). In any case, Basch's inability to use a bow is only an in-game mechanic. If Basch can't use bows, then ranged weapons have no advantage apart from allowing the wielder to attack from the back row.