Can Windsinger be played from hidden on a battlefield(1) to trigger his ability on another battlefield(2) when my opponent starts a showdown on the battlefield(2) with a unit that has 3 or less might?

No, you cannot.

According to Rule 811.1.d.2 and supported by RiftJudge FAQ #10647, when you play a card from Hidden, any targets chosen by its "When you play me" triggered ability must be at the same battlefield where that card was hidden.

Because you are playing Windsinger from a hidden state at battlefield (1), its ability is restricted to targeting units at battlefield (1). It cannot target a unit at battlefield (2).