closing a circuit programatically if ground can't be interrupted
I would use a relay for this application.  The Arduino would operate the relay, probably with the aid of an NPN transistor, and the relay contacts would be connected in parallel with the switch.

The relay provides galvanic isolation, and the contacts won't care whether they are switching AC or DC, or what polarity DC, or what voltages the contacts are relative to the Arduino ground.  (Assuming the contacts have an appropriate rating for the current they are switching.)