I'd say remove the burn after the wielder dies, and maybe lower the stability. Maybe.
People shouldn't be able to score a kill after they've already died = were outgunned.
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Making the DoT useless or require more than 2 shots would almost make it's perk useless and thus make thorn almost pointless as a exotic weapon and take a slot when compared to other exotics. As much as people dislike the DoT, it's what makes thorn relevant and unique. Maybe a stability nerf or a fire rate nerf will help it, but I think the DoT shouldn't be touched aside from removing it upon the wielders death.