It also felt troublesome to me, so here is more information to make it more consistent in my mind:
Was it cowardice?
No. She demonstrates multiple acts of bravery, always above-board and honorable conduct.
- She effectively warns Aslaug, and declares her intentions in advance.
(s4e12 20:00) We know Lagertha is true to her word. (s4e05)
- She acknowledges Aslaug's threat of vengeance by her sons with full understanding (s4e14 05:00), and braves that risk.
- She does not flinch when Ragnar's son Ubbe comes at her in a blind fury. (s4e14 20:38)
- Aslaug is not a warrior, even offering Horik's sword. Killing her in combat seems as brave as fighting a turnip.
Will the people think less of her now?
Doubtful. She used skill in melee, judicious force, alliance building.
- She trained and prepared her people until they were ready. (s4e11 09:10) She is a shieldmaiden that proves herself in combat. To use another phrase, she "pays the iron price".
- She recruited allies (Torvi), and gained support and trust that was not built on deception or bribery.
- When the outcome is decided, she calls a halt to the fighting to save lives.
- She handles Ragnar's sons before engaging battle.
Why did Lagertha do what she did?
With the arrow to Aslaug's back, it was the gentlest least humiliating.
- Aslaug deserved to die as a queen, not alone. On the road without her sons, there's worse ways to die. In fact, walking into the wild alone and defenseless seems the oddest course; I notice on re-watching that Aslaug almost exaggerates this request for the crowd to witness. She's not asking for volunteers.
- Another thought is the option against suicide, and more acceptance as in sacrifice. She is defeated, her reign is at an end. How will you choose to leave? She also believed Ivar drowned which was a strong part of her will to live.
- Sometimes the arrow you don't see is the kinder. The arrow could be the quickest deathblow with the least mutilation for her stature.
- Aslaug lost standing with her people (s4e14 24:15, s4e12 08:45), this is arguably the most dignified way for her legend to end.
- According to the tradition, to become king, in this case queen, you must kill the king (s4e10 43:20). They exchange words with the hard truth that Ragnar was dead. So only Aslaug symbolized the seat of king. You have to kill her.