websocket stuff is probably a little out of my way, buuuuuuut depending on how binding for streamdeck works - this might already be possible!
If you save a roll expression, you can then edit it (right-click/long press and then click the wrench icon) and give it a hotkey (which can be a keyboard or gamepad input). Pressing that input will trigger the expression to roll, so if the streamdeck can trigger inputs like that - it may already work?
tbh though I don't know myself for sure how a streamdeck works and I don't have one to test, so I'm mostly just typing this and hoping for the best for you!