I was looking through some photos of New Orleans and I noticed that there was an extra organ, and I recognized the faint profile of the Fender Rhodes Piano.

Better shot:

You can see Ray reaching to play the Fender Rhodes Piano while keeping his left on the Piano Bass. Ironically, the curved profile of the FR Piano probably prevented the Piano Bass from being placed upon it.

dj_vera kept bugging me to listen to Dallas to see if I could hear it, so I did.
The results (quick check, feel free to add or argue):
Love Her Madly
Gibson until "seven horses seem to be on the mark, yeah" and FRP for the first part of the break, then when it changes it goes back to the Gibson for the remainder of the song.
Backdoor Man
No FRP.
Ship of Fools
Gibson intro, FRP instrumental break, then back to the Gibson.
The Changeling
No FRP.
L.A. Woman
Tricky.
Sounds like the very beginning of the intro is the FRP, then to the Gibson.
Back into the FRP in the middle of Robby's solo at the beginning of the instrumental break. Back to the Gibson after "drivin' through your subuuuuuuuuuurbs" and for the rest of the song.
When the Music's Over
No FRP.