As others have said above it's important as the engineer to always capture a workable signal from all sources. You will be blamed and your name is on it. 
So you just learned a lesson and from now on you will make sure you record using a DI and the Mike. 
And even if it's a client paying you and they think it sounds fine, If you think it's crap then you'll have to decide if you want to carry on working with this client. 
Twice I have sent clients home as I could see a very bad sounding album happening because of musicians trying to tell me what to do.  And the issues both times were to do with them insisting on using their equipment which was in bad shape and sounded terrible. 
Most musicians are happy to use my equipment and even my Bass or guitars if need be. 
Not everyone owns recording friendly gear.
That's why all good studio's will have the commonly used gear on hand and maintain it.  Including a good drum kit, bass amp and even a good old Fender P or Jazz bass. 

Sanderxpander
I don't mind when someone wants to have "his own tone" in Sonar, and certainly I would prefer a really good guitarist to bring his own gear rather than fiddle with amp sims to create the sound that he can dial in in seconds.
However, that doesn't seem to be the case here. Without knowing more background story, I can see one of two scenarios;
1. You are paying him for recording bass on YOUR project. In which case he needs to STFU and do what you're asking, or
2. He's paying YOU to make a recording of him on HIS project. In which case you can let him listen to the recording (with the rest of the band, preferably) and say that you don't think this sound is very workable. He can either agree and redo the recording, or he won't agree and you'll deliver a crap bass sound because that's what the client wanted.

I would have to agree completely here.  I really can't speak from a recording perspective so I'm not sure what is acceptable for an engineer to expect or demand. However, in a live sound environment it is not really up to the engineer to try to change or alter the overall sound of the group.  Perhaps the band's "sound" is to sound like crap. (shrugs) Wouldn't be the first time...

Thanks for the replies. The problem is it really didn't sound bad until I tried mixing it with everything else, then it just became over saturated in some places. It will have to be re-recorded. I think if I run direct out on his amp I can capture a clean signal and he play as dirty as he wants.

Depending on the part, if you can double it with a keyboard sampled bass, it can provide an undercurrent of notes with the distortion as a layer on top. I actually do this quite a bit with bass but using parallel processing, where the direct bass sound itself is one path and the saturated sound is in parallel. That way you can combine them for just the right balance.
 
Also good tip from Dan about using the transient shaper with bass, even if it's not applicable in this situation. Hmmm...given that it's DSP month, maybe I should write an article about that.

Bflat5
Thanks for the replies. The problem is it really didn't sound bad until I tried mixing it with everything else, then it just became over saturated in some places. It will have to be re-recorded. I think if I run direct out on his amp I can capture a clean signal and he play as dirty as he wants.

I would try recording the both the DI and mic'd signal like others suggested. An SM57 will do nicely for capturing the buzz from the cab. Bracket the mic track with hi- and lopass filters, that way it can lie on top of the "meat" DI track as a special effect. The "fuzzy" sound is obviously something the player likes, so you could throw him a bone.
 
I do not agree that a paid engineer should be an apathetic puppet to the whims of the Talent. Even if your're not getting producer's royalties, your name goes in the booklet and you owe it to both yourself and the band to try and nudge their opinion towards getting the best possible result!
 
Sven

John
Why not record both with a Y connector. Mic and direct in on two tracks? 

I agree with John. You could even use the direct for the lows and the saturation for the upper mids.