Unfortunately not. The cover we provide when you take out instructor insurance with membership is served from a group cover policy. This means we allocate and pay the insurer for your place on that policy, to run alongside your membership.
Issuing a certificate of cover in a club name would unfortunately be considered misleading by the insurer, as it is just yourself as the registered / insured instructor who is insured, but the certificate in the club name could imply a legal entity (such as a business etc) is instead covered.
You are very welcome and free to trade as a club name, however. For example, John Smith joins, and is issued membership and insurance in his name. John is able to trade as a club - so 'West Sussex Martial Arts' and that's absolutely fine, but West Sussex Martial Arts is just a trading name, and not an insured entity in it's own right.
If John was to register that club as a company, charity or other legal body, he may want to consider business insurance (known as club cover). You can find out more on club cover here.