If my opponent plays Ride the Wind, can I chain a Temporal Breach on a Fizz at the battlefield to play a Hard Bargain from my trash, or would Fizz's play effect be on a separate chain?

Yes — it all happens on the same chain. Fizz's play effect doesn't start a new chain; it's added as a pending item on the existing chain, and you can use it to play Hard Bargain to counter Ride the Wind.

Here's the sequence: Ride the Wind finalizes → you react with Temporal Breach (Hidden) on top of it → Temporal Breach resolves, banishing Fizz and playing him back to the same battlefield → Fizz's "When you play me" trigger fires during that resolution and is added as a pending item on top of Temporal Breach (Rule 401.2 — the chain keeps going). Once Temporal Breach finishes, Fizz's ability finalizes, you choose Hard Bargain (energy 2 ≤ 3) from your trash, and it resolves by playing Hard Bargain onto the same chain — where Ride the Wind is still sitting below it.

Hard Bargain then resolves first (LIFO), targeting Ride the Wind. Your opponent can pay [2] to save it, otherwise it's countered. The chain only ends when everything on it clears, so Ride the Wind can't sneak through in between (Rule 330). One note: when Fizz's ability plays Hard Bargain, it ignores the energy cost but not the Power cost — Hard Bargain has none, so it's free.