Severe Punishment in damage role applies a searing aura on the player that damages nearby enemies over time. This damage over time does indeed crit and is affected by stats. For whatever reason, this DoT does not proc the Strategist Card's Tactical Advantage. Not once, not a single proc. Multiple crits, multiple tries... not even once.
Confirmed (on test server). This seems to be specific to Rage, as Violence and Galling Eruption (along with equivalents in other Tank powers) do proc Strat Card.