I am very sorry to hear this. It is really wonderful that you are thinking this way to be of benefit and a great blessing for your dog.
Reciting the mani mantra (OM MANI PADME HUM) is very effective as is reciting Amitabha's mantra (OM AMI DEVA HRIH).
You can use the practice for sentient beings at the time of death from Lama Zopa:http://www.amitabhahospice.org/public/s ... breath.pdf
None of these require empowerment.
It is best to recite audibly in the presence of your dog. There is tremendous power in all of these mantras to benefit at the time of death and the feeling of love can provide a very positive environment for his or her passing. Tonglen is definitely positive but the benefit of mantra is that it is heard by the being and forms an imprint, perhaps propelling it to a pure realm. It is really important to keep any sense of grief in check in the immediate period after death. It is said that a being in the bardo after death has senses that are seven times as acute and any strong anguish or sadness can be very upsetting and confusing to them. Whatever you can do to project a strong feeling of love and compassion will similarly be of tremendous benefit.
If you like, please post a picture of your dog with his or her name and I am sure several of us here will recite prayers and mantras for both of you from afar.