Their cleint should not avoid them to use different signature because it is included on their contract. Besides the only job of bounty manager is to handle and manage a certain project. It is not their job to help the project to advertise or endorse the project.
Yes, basically it is not the manager's job to advertise the project through his signature, but it will also give a little additional assessment to the project if a manager also advertises the project he manages in his own signature, even though it is not a special task for a manager because in terms of following a project it must be purely from the desire of his own heart, not based on coercion from others.