If I have a unit that held at a battlefield, and then I moved that unit to base, and after I had a different ready unit at my base, that I moved to that battlefield, would that cause a new showdown?

Yes, it causes a new showdown.

When you moved your first unit from the battlefield to base, the battlefield became empty and uncontrolled (Rule 323.6). When you then moved a different unit from base to that battlefield, you were moving to a location you did not currently control, which applies Contested status (Rule 445). A showdown is then opened during the cleanup after that move (Rule 344.2, updated FAQ clarification).

Since the battlefield was empty when the showdown begins, and assuming your opponent doesn't also move units there simultaneously, it opens as a non-combat showdown. You'll regain control once it closes.

Important nuance: You already scored that battlefield this turn from the Hold, so you will not score another point or trigger Conquer effects on this second showdown (FAQ #7447, Rule 465).