Yes, you can put a CAT B back on the road. Various reasons for CAT B status, not all are related to extent of damage.
You write to the DVLA and they will give you a new log book. DVLA only use a VIC marker system so the car will just have VIC mark against it. DVLA hold no details on CAT B, C, D etc. (or whatever the new ones are). They may ask for inspection but unlikely (they stopped doing this a couple of years ago)
However, it's the insurance companies that have an issue with CAT B cars so when you come to insure you may have a problem.
Not another NSX project!
Porsche 997 GT3
Porsche 981 Boxster S
BMW E46 M3
1998 NA2 Midnight Purple Targa Manual - Gone - to Thailand
1999 NA2 Kaiser Silver Manual Track Car - Gone - and found its true destiny on the race track
1992 NA1 Sebring Silver Coupe Manual - Gone (although still in my showroom in storage)