As you have to pull the epb to activate it I think your fine in that respect.
However.
as a fellow dog owner for many years I cannot emphasis the importance of getting the correct harness and seatbelt attachment and training in place BEFORE taking the puppy out in the car, especially as you say it was unsettled in the car
Afterall, you wouldn't take your children/grandchildren out without making sure they are properly strapped in
Ours have harnesses and due to patient training are no bother in the car.