Based on everything I checked, no.
Game Rant repeats the story. Bulbapedia only includes gift Mew and Jirachi and in-game trades. Seribee seems to confirm Shadow Giratina isn’t shiny-locked because it mentions all legendary Pokémon aren’t shiny-locked.
Shadow Giratina doesn’t seem to appear in the Pokédex. Couldn’t find much more about that. Hope this helps!
Edit: So YouTuber The Legendary Nic tested out what would happen if you used Transform with Mew against Shadow Giratina. The Mew ended up turning into a regular Orgin-Form Giratina. I believe this would mean that if you could possibly catch Shadow Giratina, it would enter into the Pokédex as Origin-Form Giratina.