Would it be too much to ask to somehow adapt Silence to Battlegrounds mode? Many game mechanics have some kind of counter, like reborn/divine shield/taunt, but others, like deathrattle, have almost nothing to help balance them.
Maybe some kind of neutral tier 5/6 minion that applies a debuff to targets and disables deathrattle effects during combat, or a minion that, when it kills a deathrattle target, prevents that deathrattle from triggering.