I'm probably missing something easy that I should not be missing but I can't see the problem here. Thanks!
Most SSR are for switching AC voltages only and require a zero crossing to activate conduction in addition to the input command to the SSR. Your emergency stop circuit is surely a DC circuit so a standard electromechanical relay would be the proper way to interface to the stop circuit and would not require to know or care what the DC voltage or polarity of the existing stop switch contacts are, just parallel the normally open contacts across the stop switch contacts.
So don't fool with any transistor switching arrangement where you are forced to tie the grounds of the arduino and VFD system together and know what the polarity of the switch contacts are, a simple DC relay is the correct device to interface your arduino to your VFD stop circuit. It will act just like your jumper wire experiment did.