Is the practice of Dorje Shugden hostile to the Dalai Lama's government?
NO, NOT IN THE OPINION OF THE 13TH DALAI LAMA.
From a biography sketch of Domo Geshe Rinpoche by Ursula Bernis
First Point - It used to be well known among Gelugpas the high esteem that the Thirteenth Dalai Lama had for the famous Mahasiddha of the Himalayas, Domo Geshe Rinpoche, whose original monastery in Domo or Tromo, or Chumbi Valley, is called Dungkar Gonpa.
When His Holiness the Thirteen Dalai Lama returned from India in 1912, he stopped in Tromo. A meeting took place between His Holiness and Domo Geshe Rinpoche at Kangyur Lhakan in Galinkang. It is said that His Holiness mentioned to his attendants that he expected a very special visitor one afternoon. Domo Geshe Rinpoche, who always looked like a simple monk, had prepared special delicacies to offer to His Holiness. He spent a long time in private talks with him that afternoon. In the evening, His Holiness asked his attendants if they had seen the very special person who had visited him in the afternoon. Surprised, they said they had only seen a simple monk in dirty, tattered robes. His Holiness replied, "That is too bad. I saw Je Tsong Khapa himself."
Second Point-The Dunkgar Gonpa oracle was considered the greatest of all Tibet. It was an oracle for the King Protector. The Thirteen Dalai Lama followed his advice to avert a disaster.
...
The great oracle at Dungkar Gonpa ... was very famous...
Several British and other travelers who explored Tibet in the early part of this century described the great Dungkar Gonpa oracle's activity and fame as well. In 1924 one of the English explorers reported that the oracle in Chumbi Valley, as the British called Tromo, was the greatest in Tibet at the time...
Another well-known moment of the Dungkar Gonpa oracle was in 1922 when the King Protector came spontaneously and informed (Domo) Geshe Rinpoche that there was some danger of aggression towards Tibet from Japan's invasion of a neighboring country. The Japanese armies were then in Siberia after having defeated the Russians. Japan was at the height of its aggressive expansionism, which, however, declined soon after, upon threats from the United States and when an earthquake devastated Tokyo in 1923. The Protector had advised renovating two stupas. When His Holiness the Thirteenth Dalai Lama received the message, he immediately renovated the great golden stupa at Ganden and also the Potala Palace -considered to be a stupa, filled as it was with holy objects- and praised Domo Geshe Rinpoche for having helped avert a possible national crisis.
Third Point-Is it reasonable to think that the Thirteenth Dalai Lama respected Domo Geshe Rinpoche as Tsong Khapa himself, visited Tromo, and followed the advice from King Dorje Shugden's oracle at Tromo, knowing that the Protector was hostile to the Dalai Lama's government?
To think such thing would be utter nonsense. Now, to tell such thing, as if people were devoid of memory and intelligence shows the intent of deceiving the general public, that does not have access to true information.