They are within their rights to sack him if the investigation turns up evidence he abused that producer. It might not make sense economically though.
If he did abuse that fellow and the penalty is very light or non-existent what does that say to all the other show staffers? That they too can be abused by the on-air talent? Real morale crusher there.
So if he did something they have to do something but taking the show off the air would be way too much.
On the other hand all three are all up for contract renewals so they might be perfectly happy to move on and work for another media company. Of course they would lose access to all their film and research and IP they've collected to date. All the things that have become beloved parts of the show like Stig, the celebrities setting times on the track, the video of their trips all gone.
So they would be starting from scratch in a new studio, with new a new staff and no continuity, no connection with the past. That would be a hard thing to do and they are not young men anymore.
If I were a decision maker at the BBC I'd point out that it's in everyone's best interest for the show to continue but Clarkson (if the allegations are true) has to apologize, pay a hefty fine (a portion of which goes to the staffer) and agree that he will be filmed every second of every working day (sans bathroom trips) and that any new outburst of a physical sort will generate another, higher fine.
I think that would work out well for everyone.
