Those factions have a specific text stating they arm those who represent them in the Crucible.
The solution to this would be that any faction marks, bonds, cloaks, emblems, not only change rep and marks to Crucible when wearing them, but Bungie may see that as hurting the Crucible itself, as the strikes offer more marks than a Crucible match.
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Right, it does say that. But why is this the case? Each faction has obvious interest in your PvE dealings. Why wouldn't they want to see armor and weapons to someone who is defending the world they live in? It doesn't make sense...