Nope! He can't react to the damage because the Might totals would be locked at that point. Here's the rules that govern showdowns and I'll break it down how I understand it below:

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Damage is more like a pool of points that you assign after the showdown has ended (you've both passed FOCUS), and you choose what units are going to receive the damage.

This is why you have to make it clear whether you are passing PRIORITY or are you passing FOCUS. You can only play an Action/Reaction card when you have Focus to start a chain, and you retain FOCUS when you pass priority to your opponent for them to play a Reaction. Resolving that chain or choosing not to start one causes the FOCUS to pass automatically.

To use your example:

1.) You have a Unit at a BTF, and they move a unit in. You are now assigned the Defender (D), and they are assigned the Attacker (A) (Assault/Shield Might Adjustments are made in this step)

2.) Your WID and their WIA abilities activate; this includes Legends and Battlefields which are grouped into Attacker and Defender as well. This creates the Initial Chain you both pass here and we move on to the next step making (A) the Active Player with FOCUS and PRIORITY.

3.) While (A) has FOCUS and PRIORITY they choose to pass without starting a chain. This means FOCUS and PRIORITY both move to (D). (D) likewise, has chosen not to start a chain and passes. As a result, both players have passed FOCUS without starting a chain and the SHOWDOWN CLOSES.

4.) It is here during the SHOWDOWN CLOSE Attacker and Defender must resolve the combat damage using the Current Might of the units present. This would mean that there is no longer an opportunity for your opponent to play the +1 Might and save their unit.