Copyright - Trademark/ Registered Company Query

I currently have my existing company name registered at Companies House, under which i currently trade. However, i have obtained copyright for a new company name which i intend to trade under, while keeping my registered company name the same, in order to be conduct business. So i will do a rebrand of my company which will officially be 'Name XX', Trading As 'Name B'.
I recently went to register this new trading name with Companies House, to safeguard the name in case I need it officially registered as a business. However this was refused on account that there is another business in the UK of the same name. Does this mean I cannot trade under this new name? I own the copyright to the trademark and, as the other business of the same name is in another industry there is not much likelihood of my company 'passing off' as theirs. But would like to be sure. thanks