I agree with the relay story. The only other check you could do, to cover the last .1% of risk, is to get the exact model number and go to the manufacturer's documentation that has detail to the level of the input and output signal specifications of the unit. They probably have a schematic of the input circuity. pdfs of this stuff have to be on their website.
Zero volts and shorting the input with the relay contacts are the same thing. No worries, use the relay or switch contacts or a FET solid state relay. You don't say how fast the trigger must be processed; mechanical relays are generally 10's of milliseconds).