From your description you should never apply Up and Down power at the same time. Also that motor is way past the posted spec for the EZIO2X4 relays. Perhaps a method to solve both problems. Use a single separate high capacity relay that is connected to the motor, as the manual controller would be. Then control the high capacity relay coil from one of the EZIO2X4 relays. The single high capacity relay eliminates the possibility of supplying power to both run coils on the motor at the same time. The NC contacts on the relay would control the Up direction (assuming Up hides the screen) and the NO contacts would supply the Down direction. This assumes the screen drive mechanism has some mechanical cutoff that shuts off the motor once it reaches its full Up or Down position. Sounds like it would if the manual control is a relay.
A single KPL button in the default toggle mode would be used to control the screen. Pressing the KPL button On would turn on one of the EZIO2X4 relays which energizes the high capacity relay, closing the NO contacts which would run the screen motor down. Pressing the KPL button again would turn off the EZIO2X4 relay, deenergizing the high capacity relay, driving the screen motor through the NC contacts, running the screen motor up.
Perhaps the screen manufacturer has an additional low voltage low amp controller that the EZIO2X4 relays could connect/control directly.
Radio Shack carries a relay with DPDT 10 amp contacts (275-217), more than enough to control the screen motor. The relay coil draw would not exceed the capacity of the EZIO2X4 relays. I’m sure you can find many other relays from the various online stores