No, let me clear a few things up for you…
Your switches are Momentary Switches to go up and down. You can see in the drawing that it says momentary switch. What that means, is when you press the up button it grounds out the connection between up the up wire (Red) and the common wire (white). Same for the other three buttons. These are labeled as low voltage but without taking a multimeter to them, you don’t really know what voltage they are. They could be 12V they could be 5, they could be 3.3 or ever (though I doubt it) 1.7. what you will need, is up to 3 relays or a minimum of 2 if you want to go up and down. Pardon my quick drawing but it’s late.
I substituted green for white since white wouldn’t show up. But since white is your common, green is in my drawing. When you want the screen to go down, you will turn on the down relay for a moment, simulating the pressing of the down button. When you want it to go up, you would momentarily turn on the up relay, causing the screen to go up. Now, if you use normal relay devices, when you turn them on they stay on. So you would need to have a rule somewhere in ST (smart lighting or webcore) to turn the switch off again after a couple seconds. Otherwise you might burn out your controller. Leaving it on for a couple seconds won’t hurt anything, I’m sure it can handle those could of seconds.
Now, you could also use a 2 relay device but you would still need to wire but both common wires. DO NOT connect the red to your NO on your relay and the black to the NC. That would mean that either the up or down is being pressed constantly and you will burn out your controller. If you get the device like you linked to, I don’t know that it would be able to integrate into ST. What you want to do, if you’re looking for cheap, is look at the ST_ANYTHING or the OmniThing Projects. (both clickable links) Now, these are more DIY and might not be up your alley. If you’re looking for an off the shelf device, then that would depend on what the actual voltage reading on the momentary switch is that you have now. It’s it is 12V, then you could use something like the Quibino dual channel z-wave relay. (clickable link). They call it the Flush shutter. Or if you are going to power it from 12-24V, the Flush SHutter DC.
However…as the name suggested, this device can control your shade (screen) but itself, without your current controller and has the option of a manual switch. So, if you’re going to spend the money anyway, it might be easier to just replace your controller, rather than trying to rig something to it. Personally, if you’re not going to go the DIY route, I would just replace it with a Shade/Shutter controller.