Absolutely!
You say "no fault", what does this mean exactly? Did you just crash into some street furniture on the roundabout? "Was hit" implies some one else caused the accident, so the problem is down to their insurance company.
You do not have to accept their or even your own insurance company's word that it is "unroadworthy". It may be unroadworthy until it is repaired but once repaired to a safe condition it is no longer unroadworthy.
And this applies to whatever happened before. So having to fix it twice means nothing at all.
I get the feeling they are trying to tell you it is "uneconomical to repair" which just means they don't want to pay for the repair as that is more than the write off value.
In the past I have bought insurance write offs back from the insurance company, used the write off money to repair them and then just carried on driving them.
Remember it is YOUR vehicle and they cannot dictate to you what THEY want done with it.
To get it declared unroadworthy even once it is repaired would involve the Dept of Transport, police inspections etc etc.
There are public bodies you can consult like the insurance ombudsman, although they tend to be run by insurance companies, or simply getting involved with a solicitor who knows motoring law. But first of all I would get an independent inspection from a qualified vehicle inspector/accident examiner. We once had to do that, he fell about laughing when he had to say to us, "so where is the damage then" as it was so slight he couldn't even see it.
For it to remain seriously unroadworthy the integrity of the chassis would have to be seriously compromised.
I'll put up a link to a thread I put up ages ago about all this an article written by a bloke who used to write for Land Rover Monthly, about how even if it is beyond "economical repair" the at fault drivers insurance can be made to pay for it to be repaired.
