How can I make a fine tipped object emit an electric field similar to that of a human finger?
Some data as what the excitation signals for the screens are would help. But even without that you could design a powered stylus with a fine tip.

The basic setup would be a simple metal wire connected with one end to a voltage source. The other end of voltage source should be insulated. Now you have a artificially generated electric field around the wire. The nice thing about it is that the tips always get the highest fields. You should put an insulation around the wire tip to prevent scratches. If you put a battery in, it won't even be drained because there would be no significant current path.

As for the needed voltage, I can't guess that well without any data. The higher the voltage the higher the field will be, what means that you can have a smaller tip.

The problem with this approach could be that the electric field projected from the tip is too weirdly distributed. I have no idea what the expected input signals for the processing system are. So it could be possible that the tip is not recognized because it has a too weird field distribution. If you knew the exact parameters of the system you could do something similar to a microatennae which gives the same inputs to the system as a fingertip.