Can you as a reaction to a unit initiating a showdown use Heedless Resurrection as a defender and play another unit to that battlefield?

Yes, as a defender, you can use Heedless Resurrection in reaction to a unit initiating a showdown to play a unit to that same battlefield.

According to RiftJudge FAQ #9753 and the related mechanics for showdowns:

1. Reaction Timing: When an opponent initiates a showdown by moving a unit to your battlefield, the showdown is staged and the state becomes Closed once the initial chain is created (which happens if there are any "When I Attack" or "When I Defend" triggers). Even if no triggers exist to open a chain, if you hold priority as the defender, you can play a [Reaction] to an opponent's action to initiate a chain.
2. Battlefield Control: During a showdown, you retain control of the battlefield you are defending. Because Heedless Resurrection creates a chain (making the state Closed), you do not lose control of the battlefield during its resolution, even if you kill your last unit there to pay the cost (Rule 187.4.c and Rule 461.5).
3. Playing the Resurrected Unit: Because you retain control of the battlefield, it remains a valid location to play the unit resurrected by Heedless Resurrection. Once played, this unit will enter the battlefield, and if the showdown is still ongoing, it will participate as a defender (Rule 459.2.b.4.a).

Sequence Summary:

1. Showdown Initiation: An opponent moves a unit to your battlefield, triggering a showdown.
2. Reaction: You play Heedless Resurrection as a [Reaction]. This opens a chain and puts the game in a Closed state.
3. Cost: You kill a friendly unit at the battlefield to pay the additional cost. You do not lose control of the battlefield because the game is in a Closed state (Rule 187.4.c).
4. Resolution: Heedless Resurrection resolves, and you play the resurrected unit to the battlefield.
5. Combat Cleanup/Designation: Following the chain resolution, the game performs a cleanup, and your new unit gains the "Defender" designation if the showdown is ongoing (Rule 459.2.b.4.a).

This is my interpretation based on the provided FAQs and the "Unleashed" rules update regarding battlefield control. No official FAQ exists for this exact scenario beyond the interaction described in #9753.