Would a salt bridge work through which the ions could not diffuse into the half cells?
It is a very good thought experiment as to why we don't cap the salt bridge ends with gold plugs. By capping, you are essentially providing an interface which can source and sink for electrons in the cells, thus each end will act as an electrode in the cell. One end of the salt bridge will act as a cathode and one end will act as an anode (strange as it may sound!)

As to your comment above "I want them to be non porous and able to let the electric field through that the accumulating charges have. I think a closed glass tube would also work". No a close tube will not work. You need the ions or electrons to move in order to observe current flow. A closed glass tube will have an infinite resistance and hence no current can flow!