The pianist you are looking for is probably a local bar pianist who agreed to provide some background music for this piano "dolly-machine" commercial. He or she is trying to emulate stride piano. It is an off-the-cuff arrangement that is most likely not written down. Your fake-book score above is what everyone starts with. Most people end up sounding like restaurant piano (like the guy/gal in your op), but often it is best to listen and strive for the source - Teddy, or Art, or Oscar, etc.
Hope this helps!
p.s. I will be very surprised if the company emails you back and says this is some famous piano legend playing the score to the commercial. I guess I would have to eat my words then.
Thanks for your reply and video, above:
I am inlined to believe that this performance just might be by a local bar room pianist (as you mentioned), although I am still waiting on a reply from the company that posted the video.
I posted the lead sheet for those folks who might not be familiar with the "Body and Soul" piece, and, I think that the performance is very well played even if it is not by Art / Oscar / Teddy / or, etc.
Will be considering doing a transcription of this particular arrangement (since it is a very good one) as no score obviously exists, for it.
Will be curious to see what the company says about it, if/when they get back to me.