If I move a unit into a contested battlefield and activate a move trigger, can I reaction speed a spell to move out of the battlefield before combat starts?

Yes — if the move trigger opens a chain, you can play a Reaction-speed spell to move the unit out before the showdown starts, and the combat won't happen.

The move trigger creates a chain (Neutral Closed State), and during a Closed State only Reaction spells/abilities can be played (Rule 813). That window is your only chance: once the chain empties, a Cleanup runs and the staged Combat opens immediately with no priority for non-Reaction play (FAQ #1645, #3814).

If your Reaction moves the unit away, the Combat ceases being Staged (Rule 323.10 / 461.2) and Contested is removed in the next Cleanup since you no longer have units there (Rule 323.11) — so no Showdown or Combat ever begins. Two caveats: your move trigger still resolves, but any "here"-dependent instructions on it will fail since you're gone (Rule 359.3.e.12), and if the unit has no move trigger, there's no chain at all and you get no window to react before combat starts.