If I kill a unit with a spell during a showdown, as the attacker, can I play Immortal Phoenix from my trash to that battlefield?

No, you cannot play Immortal Phoenix from your trash to the battlefield where the showdown is taking place if you are the attacker and do not control that battlefield.

The Ruling

According to RiftJudge FAQ #4527:

"Ruling: You can play Immortal Phoenix from trash to your base or to a battlefield you control (even when defending). You cannot play it to a battlefield you don't control (so not when attacking) or to an open battlefield."

Why this is the case

1. Battlefield Control: When you initiate an attack, you are moving units to a battlefield currently controlled by your opponent. Therefore, you do not control that battlefield.
2. Valid Destinations: Immortal Phoenix's ability allows you to play it from your trash only to your base or to a battlefield you control. Because you do not control the battlefield you are attacking, it is an invalid destination.
3. Defending vs. Attacking: While you can play the Phoenix to a battlefield you control when you are the defender (because you own/control that battlefield), the restriction specifically prohibits playing it to a battlefield you do not control, which is the case when you are attacking.

This interpretation is based on the official ruling provided in RiftJudge FAQ #4527.