If I deal 1 to a 3 might unit in base with Bellows Breath and that resolves, then I play Thousand-Tailed Watcher, can the opponent react to play Discipline to bring it to 5 might before it drops to lethal for the unit to die?

Yes — they can, and the unit survives.

Discipline is a [Reaction] spell, so it's legal in the Closed State created when Thousand-Tailed Watcher's "when you play me" ability goes on the chain (Rule 338.1.a.2). Because it's added on top, Discipline resolves first: your opponent's 3 Might unit becomes 5 Might, they draw 1, then Watcher's ability resolves and applies -3, leaving it at 2 Might.

The 1 damage from Bellows Breath is still marked on it, but damage doesn't reduce Might — the unit only dies in cleanup if marked damage equals or exceeds its current Might (Rule 323.5). 1 < 2, so it lives. Note the -3 is snapshotted at 3 (not capped by the "minimum of 1" clause here, since 5 - 3 = 2 is already above 1), so any further pump still works normally.