I don't remember the source, it may be in one of the home movie clips at YouTube, or buried in Memory Alpha, or possibly from one of Shatner's Memories book. I've been reading about Star Trek for 47 years, so I'm sorry that I can't be more precise.
At Memory Alpha they state he did temp voices on the show (Trelayne's parents) that would later be dubbed over by voice actors. I have no doubt he did the voices on set, but I doubt he went back later to dub the rabbit.
Trek constantly used veteran voice over actors to do the voice-overs....
Vic Perrin----Outer Limits
Robert Johnson---Mission Impossible
Ted Cassidy
Walker Edmiston---Keebler elf
Bart LaRue
Elizabeth Palmer
Malachi Throne
Abraham Soafer
Later they used James Doohan who was going to be looping dialogue anyway----- a lot.
It's also said that probably Grant Woods who played Kelowitz in Arena--looped in the faked Commodor's voice in the same episode.
I really doubt they dragged in Blackburn for the looping session for the episode----he never spoke a line of dialogue in any production he ever appeared in.
http://en.memory-alpha.org/wiki/William_Blackburn