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.