For the first option, yes, if you bring your own payout partner (Bank or MTO), the FX gains are yours.
May I humbly request you watch these two videos? They explain the concept of what correspondent tie-up agreements are – very well.
- What is a Tri-Party Correspondent Tie-Up Agreement
- What is a Correspondent Tie-up? & Why is it necessary?
As far as the second option is concerned, the FX handling is absent on the US (load) side and is only applicable on the payout side. And this matters on how you have positioned it, whether you use your own payout partner or not.
The sponsor only deals with USD.