The shop on the door is William Peat. The website opens as Castle Bank.
What I saw
The two shopfronts at 25 and 29 Horsemarket trade as William Peat Butchers and William Peat Deli & Bakery. The signage, the Q Guild listing, the Teesdale Mercury feature, the Instagram handle @williampeatbutchers, all under the William Peat name. The website lives at castlebankbutchers.co.uk and opens with a wide red-banner H1 saying "Meet your trusted North East family butchers" attached to a different brand. A customer Googling "William Peat Butchers Barnard Castle" lands on a Castle Bank corporate page that mentions William Peat by name three clicks deep.
What the rebuild does about it
The rebuild speaks as William Peat. The Horsemarket address, the centenary, the founder's name and the Q Guild standing are above the fold on the home page. The Castle Bank operating relationship is acknowledged in the heritage block where it belongs (1993 onwards), not as the brand that greets the visitor.