It wouldn't really do anything useful for a Trailer in itself. It's not his big signature song people look for. It needs to be one of his songs for the game(whether he gets a new stage or not, who knows. Radical Highway and Westopolis are the best options he has due to recognizability). All Hail Shadow and I Am (All Of Me) are his big ones, to the point they both have full out OST releases. Radical Highway is cool, but don't actually represent Shadow's characterization at all. Songs without lyrics rarely can do this, especially considering signature songs have lyrics in the Dreamcast saga. The only song from Adventure 2 that is close to a signature one is oddly enough none of the stage songs, but his main theme, Throw It All Away(eep, the edge in that song. Yeish), and that doesn't represent what Shadow is anymore. He's an anti-hero. Even Sonic's trailer theme was a lyrical one. And for good reasons. It sets the stage for awesomeness. Radical Highway is instrumental, and again, while good, doesn't really show off his outright characterization.
His theme should match his characterization first, and people remember Shadow now because of his edge(though honestly, there are people who only remember his due to his guns... which while they are notable to his character now, aren't his most iconic abilities. And don't have to be in. Referenced as you noted with the super scope, but hardly a necessity. I'd love them in myself, but they were not all that liked compared to his Chaos moves). The most important part to the trailer itself is how to handle it. All Hail Shadow might not work if Shadow is basically being violent in general. It's his Hero theme. So unless he's killing a villain/antagonist, it would clash a bit. I Am (All Of Me) might work better. The choreography is also an issue. It has to all fit together. Though him killing, say, Bowser Jr., getting the splash screen, then the music starts up works too. I Am (All Of Me) might work better due to the slower start of the song. All Hail Shadow might only work for a trailer with the remix. That small intro makes all the difference.
If you're wondering why the choreography matters to me, I'm a huge fan of presentation and have watched stuff like Power Rangers. Timing is everything. When stuff like the Ultrazord appears, for instance, it works because the music ties into it perfectly. Whether it's a strong riff that tells you stuff is about to happen, or exactly how Titanus rolls in while the music is playing. Even the bad Disney reimagining got this right, which Titanus' name showing up with lightning sound effects without drowning out the music. This matters a lot. I probably care too much about it, but these are cinematic trailers, so obviously they want them to get the most hype.