This fight sucks if you don't have any long range, it's exactly for fights like this when you need them the most because archer is the must troop to kill heavy armors but if u have no long range of ur own, enemy cats will just eat ur archers 1.5 every round.
Ok so if I were you woud take 7 archers 1 legioner or soldier. Keep the legioner/soldier as your most forward troop every time when cats start shooting. In ideal I would actually take a 9 hit point legioner/soldier to the fight. Anyways if it stays as the most forward troop it will take usually at least some catapult hits into himself for low damage so ur archers stay alive longer. Then I would just rush forward and finish the fight quickly before those catapults decimate my archers.
Edit: I assume this the fight you meant? It's the top sector in sturm and has all the units you said plus also one mercanary. Anyways this is with 7 archer 1 legioner:
High losses, but still you should win 8 times out of 10.
And as comparison with 4 archers 4 ballista:
Much better losses and you win 10 times out of 10.