Wikipedia tells me ' a phoenix is a long-lived bird that is cyclically regenerated or reborn. Associated with the sun, a phoenix obtains new life by arising from the ashes of its predecessor'.
In your case Cooper is the new pup's predecessor, his passing having left a space for another dobie.
Phoenix is also associated with long life. Here's a lovely poem called 'The Phoenix who survived' by Dean Fraser-Phillips:
Rising from the ashes, you spread your wings tofly
Reaching now for those final dreams,belief will never die
Beautiful andsoglorious, the eternal bird offire
Soaringup intothe sun,leaving behind the pyre
Dreams are now your destiny;reach with all your might
You’re a powerful inferno,ablaze andinfull flight
The sky is now your playground your home is now the sun
Destiny isinyour grasp,forgetthe doubts and fears
Your flame will burneternallyfor all remaining years
There will be times of doubt, andmomentsofpure sorrow
But one thing thatisever true, you’re the angel oftomorrow Poem - The Phoenix who survived - Deanthebard | Deanthebard