Technically the FOMC chair is different than the chair of the Board of Governors. Jerome Powell holds both positions.
The FOMC chair is chosen by the Board of Governors.
The president nominates the members of the Board of Governors and the chair of the Board of Governors.
Technically we could have a different FOMC chair and chair of the Board of Governors. But doubt that the Fed would go that route.
The Senate does push back on Fed nominations. Notably one of Trump's crazy Fed picks didn't get confirmed by the Senate during his first term.
But I feel like the Republican Senate caucus is quite homogeneous these days. I can only think of a handful of Republican Senators that meaningfully defect from the caucus anymore.