This perk's effects still apply with Dogmeat as an active companion.
It seems that the player character needs to fast travel or wait a certain amount of time for the effects of the Lone Wanderer perk to take effect (i.e. for damage values to change) after a companion is dismissed.
This perk and Attack Dog are unique in offering actual damage reduction instead of a set amount of damage resistance. It offers an increasing amount of effective damage resistance depending on the player's damage resistance.