No because it wouldn't work for Sei and Build outside of the heavy use of Plot Armor.
If we take the suit and put it in the battle royal scenario with real people and not mindless Death Drones then things become much harrier.
In fact Build Fighters does address this very thing, just not with Build Strike. In the Battle Royal we see Build Strike getting its ass kicked by Epyon, it wasn't really able to stop it but because of Epyon being Melee only it was simple to shoot it down the moment it lost all it's support.
But again, Domon did well in the Battle Royale as I recall. Even after it got hijacked by the Devil Gundam. And as I recall there's another episode where the Dynames gets taken down even though it has a sniper rifle, so I don't think a gun is a make or break thing.
Besides, it's not like the God Gundam doesn't have projectiles. Sekiha Tenkoken and all that, but more importantly, I think, is that having a gun goes against the entire style of the Shining and God. That's not what they're about. Imagine if the Maxter had a sword. It's not their fighting style. If you want a Gundam that uses guns, get something that isn't primarily built with hand-to-hand combat in mind.
EDIT: But I mean, I guess if they can make it cool, whatever. Just doesn't really fit the suit's motif or purpose.