If the Runner is tagged, Dedicated Response Team gains "Whenever a successful run ends, do 2 meat damage."
They don't call them dedicated for nothing.
Because it is the Runner's turn, the Runner must make another run with Doppelgänger before Dedicated Response Team can resolve. The Dedicated Response Team trigger must wait until the second run is completed before resolving.:Ruling
The runner makes a successful run. At the end of this run, both Doppelgänger and Dedicated Response Team trigger. Doppelgänger must trigger first, because the runner is the active player, and the runner chooses to make another run. When will Dedicated Response Team deal damage? You are correct that the Runner’s Doppelgänger would trigger first, and the run must be resolved before the Dedicated Response Team. So, effectively, if that run is also successful, the Runner could be taking 4 meat damage. Or, perhaps, the Doppelgänger run trashes the Dedicated Response Team, and the Runner will take no meat damage!