Well, according to c-man's notes, the engineers present for each Pet Sounds song's final session who would've likely handled the mixes with Brian were:
Sloop John B - Chuck Britz & Winston Wong
You Still Believe in Me - unknown
Pet Sounds - Chuck Britz
Let's Go Away for Awhile - Chuck Britz
Wouldn't It Be Nice - Ralph Valentin & mysterious Don T.
Caroline No - Chuck Britz
I Know There's an Answer - possibly Bowen David
Don't Talk - Bowen David
I Just Wasn't Made for These Times - Ralph Valentin
That's Not Me - unknown
I'm Waiting for the Day - Ralph Valentin & Don T.
God Only Knows - Ralph Valentin
Here Today - possibly Jerry Hockman
A couple of early Wouldn't It Be Nice mono mixes from Feb 16 and March 3 would've been Chuck at Western. Lots of early mixes going around but I think that's the only song with totally different vocal attempts at different studios.
Yeah, and don't forget that Jim Lockert and Phil Kaye reportedly engineered some
Pet Sounds sessions, according to Byron Preiss. Whether or not that means mixing or simply vocal overdub sessions, we don't know.