If I am defending with a single unit in a showdown with an enemy unit and have a Resonating Strike hidden, can I flash back to base, then cast Resonating Strike back to the battlefield while the showdown is still ongoing, even though it was the only unit there?

Yes — you can Flash your unit to base and then play the hidden Resonating Strike to move it back into the ongoing showdown.

Two things make this work. First, you don't lose control of the battlefield while the showdown is ongoing (187.4.c; the corrupted-control cleanup only applies in an Open state with no Showdown or Combat there), so it's still "a battlefield you control" for Resonating Strike's destination. Second, the Hidden restriction on targeting (811.1.d.2) doesn't block it: the battlefield choice must be the one it was hidden at, but "a unit you control at a different location" can never be satisfied by a unit at that same battlefield, so that target may be chosen freely — including from your base.

When the unit arrives back, it gains the Defender designation in the cleanup following the action that placed it there (466.2.c.3.a), and it gets +2 [M] this turn. Just remember you need priority/Focus to play Flash in the first place, and playing it only passes Focus rather than ending the showdown.