That relay module looks eminently suitable.
When you assemble this into a box, you want a cable with the 6 pin DIN plug for the projector, and also a matching 6-pin socket on the box, so that the original controller plugs in to the socket.
Essentially all connections are wired straight through from the lead with the plug, to the socket and your relay contacts simply parallel the two buttons. There may be slight adaptations of this wiring if there is a concern about interference from the hand controller, but that is probably not likely.
I seem to recall having a couple of these controllers (but I don't recall the projector itself) in my archives. 