I expect Stream Deck is just sending keys to an application with using SendInput or similar? That doesn't work. Trainers and Aurora listen for virtual key presses application wide. To make this work Stream Deck would have to trigger a key press on hardware level (or via a driver) which it probably cannot.
I don't have the Stream Deck but can you run commands from it? Just like on Windows Win+R? Can you enter URLs there and if you then click on your button does it open the webbrowser? If so I could add a file protocol that lets you trigger a cheat.
A file protocol is an URL which sends a command to an installed application instead. Just like for steam for example: steam://install/8230
I tried on mine and that is possible. It has a "website" button option in the "systems" group. That lets you enter a url that then opens in the default browser when pressed. It also has a selectable option for "Get request in background"
Here is a link to what it looks like in the Stream Deck software Link
I also attached the file to this post.