The entire route is ridable in its entirety. I know I've ridden it dozens of times as I live mid way along it. There was even a time when i thought riding it was some kind of endurance challenge!
The sandy bits at the northern end are challenging but any halfway competant mtb-er can handle them easily if they have a semblance of a sand riding technique and can spin like a bandit.
I've ridden it on my mtb, my 38mm tyred roughstuff tourer, and regularly use the stretches north and south of horsham on my 28mm tyred audax bike. boy does it make the mudguards rattle. but other than that it is no big deal you just have to go slower on thinner tyres. A CX bike would laugh at it a croix de fer would spit in its eye.
last month did bramley to shoreham in company of someone on a
condor road bike with 25's on. don't think he enjoyed it as much as he could have done/I did but he rode the whole thing and was still walking and talking at the pub afterwards and rode on to Brighton.
there is very little tarmac btw, and then mainly on the short sections where it goes on road (we are campaigning locally to get these diverted)