Redmine

Could this just be done as a new user role that would have special
attributes if it
were assigned to someone on a project? E.g. anyone with the
role Client on a
project would have normal Reporter status but also show in the
Project Overview
page?

You can create a 'Client' role and give to this role the same
permissions as the 'Reporter' role.
On the project overview page, user with this role will be shown
in the members area as the 'Client'.