If I use the Akali legend ability to move the red Akali champion back and it ends a showdown, would the defending units heal before the move trigger resolves?

No, the defending units do not heal before Akali's "When I move" trigger resolves.

Here's the sequence:

1. Akali Legend ability resolves → Akali, Deadly Weapon moves from the contested battlefield to your base.
2. This makes her "When I move" ability Pending, and a normal Cleanup runs (Rule 448 — "When a Move action is complete, perform a Cleanup").
3. During that Cleanup, the combat/showdown ceases to be staged — the attacker's only unit left, so there are no longer units from two opposing players present (Rule 323.9.a). No Combat is staged.
4. No healing occurs during this Cleanup. Healing only happens during a Combat Cleanup (which requires the Combat Damage Step to be reached — Rule 461.1.a.1) or at end of turn (Rule 143.3.b). Since the attacker's unit left before the Showdown closed and Combat Damage was dealt, those steps are never entered.
5. After the Cleanup, FEPR processes the Pending trigger. Akali's "When I move" resolves, letting you deal 1 (or 2 if Empowered) to a unit at the battlefield she left.

The defending units are still damaged (if they had any) and can be targeted by Akali's trigger. There is no heal window between the move and the trigger's resolution.