Specific impulse of exhaust velocity depends on the temperature in the combustion chamber. Won't regenerative cooling lower the chamber temperature, causing the exhaust velocity to decrease?
In a regeneratively cooled engine the heat extracted from the chamber and/or nozzle walls is absorbed by the fuel, which is later burned. So compared to e.g. an ablatively cooled engine, the reactants start combustion at a higher temperature and therefore finish combustion at a higher temperature, with more total energy available for conversion to velocity. Therefore regenerative cooling actually increases Isp compared to other cooling methods, by recapturing some of the energy that would otherwise be lost (radiated or ablated away) and returning it to where it is wanted.