1: If they burn elewen, they also burn any operation Elewen was connected to.
2: "Objective proof that one of our allies is actively betraying us" is much more actionable than a basic suspicion. Kind of thing people back in the capital should know about. Kind of thing that takes president over what's basically a petty squabble.
3: It's also proof that Ulfric is being manipulated, which may make Ulfric take pause and reconsider his actions.
Play your cards right and bringing up that Dossier gets the Imperials and the Stormcloaks pointed at the Thalmor instead of each other, and opens up the possibility for a more proper peaceful resolution to the war in favor of screwing over the damn elves.