how to get this Fender Rhodes sound?

That was the short answer. Personally, I only like distortion part of the time, and didn't really care for all the break-up in that video. Plus, read with a grain of salt, because I am mostly a guitar guy. But, my wife plays keys, and while recording her (trying to get a clean sound), I routed her Kurzweil through my Digidesign Eleven rack (in error). Its preset was for massive distortion, and the keyboard sounds were just that. So the long answer would be, yes, use a tube amp (I have a high-gain THD 15 watt Univalve that is very hard to get clean) if you have one. But if you don't, run the keyboard into a distortion plug-in. I would add that I have tube amps that do things impossible to do with modeling (IMHO), but you could certainly try to get close. The Rhodes has its own strain of break-up when you whack those metal tines, but it sounded to me like it was the amp that was adding the flavor your client's ears demand. If the client likes that sound, ask him (or her, selon le cas) to bring in the guitarist's pedals and amps. I would be surprised if they are using solid state, but maybe. Anyway, any guitarist without a tube amp must certainly have an overdrive pedal, and some of those can come close to producing output tube breakup.

What if you want the soft notes to be clean and the loud ones to be distorted? Maybe record two audio tracks and use automation to fade in the loud passages with the added distortion. I imagine you could also rig up one track with a limiter, and one without, thereby using a side-chain with its threshold set to trigger the distortion once the keyboard hits a certain dynamic level.

But, getting back to my initial reaction to the video. It just sounds like the Rhodes (or rather, the emulation of the Rhodes...) is plugged into a tube amp.
lovely and informative post, Juan - you have really done me a useful service - thanks for taking the time.
As the guy who is asking me to replicate someone else's sound is himself a guitarist, maybe I should I ask him how long it took him to develop his own sound - and could he emulate someone else's at the drop of a fader? :)
Having said that - my Fender keyboard playing has very much taken a massive back step as I haven't used a real fender for many years - I set mine up myself and it took a lot of skill and care. Since then I have gone digital and while my Piano, the Kawai MP9500 is excellent as a piano, it's pretty weak as a fender emulation, so I haven't really developed my digital fender sound as much as I should.

Your comments are stimulating more research :)

I love the Rhodes, it's a sound I really miss on records, especially jazz records. It's so easy to get a great sound, I'd never use a simulation at this point.The Fender Rhodes electric piano possesses one of the most recognizable sounds in modern music.The classic Rhodes sound is highly expressive part bell, part xylophone, and part piano.The Rhodes electric piano is back in production and better than ever.