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).
"Twenty years from now you will be more disappointed by the things that you didn't do than by the ones you did do. So throw off the bowlines. Sail away from the safe harbor. Catch the trade winds in your sails. Explore. Dream. Discover." -- Mark Twain