If you have not settled with your insurance carrier that has categorized your vehicle as a Category B salvage or write off then you may speak with the adjuster that is working on the claim to see if you can appeal this decision. The insurance company should at least provide you with information on how they determined the vehicle to be a Category B instead of Category C salvage if there is not structural damage as you noted.
If your company does allow an appeal then you will likely need to get information supporting your claim that there is not structural damage and that the car could be repaired so that it would again be roadworthy.
As you are aware a Category B write off basically means that the vehicle must be broken for spares. These vehicles must not be repaired. These vehicles should only be sold to licensed and approved vehicle dismantlers.
With a Category B the vehicle will typically have been damaged beyond economical repair, usually with major structural damage. The DVLA will require "Notification of Destruction" but parts can be removed and sold on.
It would appear that you want your car to be instead written off as a Category C which ensures the vehicle itself can be repaired. Keep in mind that sometimes repairing such a car may exceed the pre-accident (PAV) value.
So Category C vehicles may be repaired although they it is uneconomic for the insurance company to have it repaired. Category C cars require a Vehicle Identity Check (VIC).
The Vehicle Identity Check (VIC) is an initiative started by the Vehicle and Operator Services Agency (VOSA). This scheme has been developed to help fight car crime by deterring criminals from stealing cars and passing them off as repaired accident damage ones.
The DVLA may also be able to advise you if you are able to appeal what category write off your insurer has determined your car should be placed in. For a UK auto insurance quote, click here.